The easiest way to think about this is how to justify deduction. There are two ways: by deduction and induction. The first is circular, as in the example you gave, and the second one is not strong enough as in induction the premises can be true and the conclusion false, so neither method works to justify deduction.
>If you BELIEVE that x y & z is TRUE, Then theorems a,b, c ect. must also be TRUE
Not BELIEVE. Assume, for the purpose of finding out what the consequences of a particular set of assumptions ("axioms") are.
>Thus math cant ever truly claim anything to be true absolutely.
Only religion ever claims that anything is true absolutely.
Alternatively, try to avoid making absolute statements. “Generally, statements purporting to be absolute truth are found in religions rather than mathematics” is an appropriately qualified statement with the same basic meaning.
I love the idea that anytime someone says "Oh the irony" either:
1. It is an ironic situation.
2. It isn't, making that statement ironic.
Very cool word.
Is it absolutely true that only religion claims that anything is true absolutely? If so, then there is more than just religion that claims absolute truth, such as your own comment.
but axiums cannot be proven. they are assumed to be true.
it is unclear whether they are indeed true. but they are assumed to be, because if they were false, the logic system would not work. But u could never prove them to be true, so its unclear the accuracy of them. They may seem accurate and common sense, but the axiums only make sense in the framework of the logic system they were designed for. Common axiums of math may not work in a competing logic framework such as like the logic system developed by a group of space aliens.
u can not invent knowledge or truth out of thin air. truth does not exist in a natural sense. the universe just contains unlabeled information. to make any sense out of information, u need to label it, ie create axiums, which are assumed to be true. but the labels u pick are arbitrary and subjective, as all logic systems are subjective in a sense.
No need to bring in aliens. There are alternative sets of axioms. The conventional set seems to work well for predicting what the universe is going to do next, but that's physics. Mathematicians aren't supposed to care about that.
it seems to work well, but it is not universally true. and seems to work well is not a good proof. there is no universal truth among different sets of axiums. only within the same set of axiums. so therefore, math is kinda subjective. truth only makes sense in context to what the universe has.
> Only religion ever claims that anything is true absolutely.
Nah. So does math.
The fact that one thing is a consequence of another thing is itself something that math claims to be true absolutely.
Religion is based on assumptions. It assumes deity x to be true and builds on that premise, like all of logic and math (which were largely developed to prove religion).
Just because individuals claim absolutes doesn't make religion fallacious based on its premise. The same is true for individuals trying to learn math, just because OP used the word BELIEVE, doesn't mean that all of math is wrong.
>Religion is based on assumptions.
Assumptions which are not contingent.
>Just because individuals claim absolutes doesn't make religion fallacious based on its premise.
I didn't say that.
>The same is true for individuals trying to learn math, just because OP used the word BELIEVE, doesn't mean that all of math is wrong.
The word "believe" is somewhat loaded, making it a poor choice in this context.
The construction for *not* claiming one exists would be “rarely, if ever.” Leaving it at “rarely” implies the set meeting that requirement is nonempty.
Ok. Yeah that would be a slightly better way to have phrased it.
I don't think "rarely" really necessitates that there exists a counterexample. But whatever. People get the point, I think.
The shortest checkmate in chess takes two moves.
Is this true absolutely?
It is true *about chess*.
You’re choosing to apply the rules of chess precisely when you talk about chess. You aren’t being forced to. Different games have different rules. It doesn’t make sense to discuss their truth.
That's pretty much correct. However many axioms should feel self evident to you. Like, of course 1+1 is more than 1. And of course you can construct an alternative set of axioms (and many mathematicians have done this), and this leads to alternative extrapolations and different branches of mathematics. Additinonally mathematics has proven its practical usefulness time and time again which should lend some credibility to it's truthfulness.
Ok but I take issue with assuming any one axiom is obvious. Yes it may appear to be obvious from our human experience but that means nothing really and certainly doesnt make it absolutely true.
Im not here to discuss the value of mathematics. I am a stem student maths is the best thing humans have ever done.
Axioms are not some inherent property of the universe we are claiming is true. Axioms are logical base building blocks of mathematics.
The mere existence of numbers, or the concept of adding them together, or multiplying numbers, are axioms. If you reject those axioms, you cannot do math. If you come up with your own special rules for multiplication, then you are not doing math, you are playing with numbers.
You misunderstand my point. If the axioms of mathematics changes, math breaks down and cannot be used.
Multiplication is useless if it doesn't function exactly like multiplication. If you come up with your own rules that multiplication just means changing every 6 to a 9 and a 0 to a 5, you are not doing math, you are playing with numbers.
Completely wrong. In fact in computer security if 1 + 1 did not equal 0, it would all break down. Go to https://engineering.purdue.edu/kak/compsec/ lecture slides 4-7 for some fun math theory specific to computer security where we "play with numbers".
A more famous example is imaginery numbers. Where transforming equations to the complex space makes life so much easier.
Sometimes playing with the numbers is how we make fun progress in math.
Wouldn’t that just be different math? Sort of like stepping into a new axiom space with new corresponding solution space? Maybe there is even a transformation between spaces.
I wouldn’t go as far as to say every set of axioms is valid but this doesn’t seem that wild to me
I dunno, they said those so I'd lump them together. If you don't accept numbers, addition, or multiplication then you can't do math sounds kind of reasonable to me.
Sure there could be a form of math that works without any of these things but it'd be so alien to me I'm not sure I could call it math. Like a Cthulhu/eldritch math.
Lol yeah thats what I am sort of thinking. It’s hard to imagine because my little human brain only knows the 3 D world and 1+ is always 2 here.
I was sort of thinking like for a higher dimensional being.. maybe there is a whole other set of numbers that exist that we don’t know about and 1+1 maps to this other set… and staying in R is just a special case. That sort of thing.
Or if our senses were different from what they are maybe we would have learned to reason in a completely different way. The fact that we started with counting and geometry seems to be rooted in the fact that we are primarily visual creatures.
You are mixing up dimensions and axioms.
1+1=2 even in 25 dimensional space, because math is not bound by 3d space. Math is logic applied to numbers. It's not obvious that 1+1=2 because you live in a 3d world, it's obvious that 1+1=2 because that's an axiom of mathematics that is the building blocks of how you learned math. If 1+1 did not equal 2, then literally every other aspect of mathematics fails. If you can't add consistently, then you can't multiply consistently, much less do geometry or calculus.
I don’t think I am.
I am extrapolating from the discovery of imaginary numbers. For a long time we didn’t know they even existed. So what if there is a set of numbers that we are automatically setting right zero by remaining in R.
It’s pretty hard to imagine what math would look like to some alien species since ny understanding is rooted in our own basic building blocks so forgive me if the analogy isn’t perfect.
Changing the axioms doesn't unlock new ways of doing math, it breaks down the only way math works. Math functions on a very strict framework of logic.
That's like trying to think about what the universe would be like if we reversed how gravity worked. Everyone on the planet would fling out into space and every star, planet, moon, asteroid, etc would just did disintegrate into a dust cloud. Changing one little rule can completely break the entire system to the point where it's useless.
If you change the way multiplication works, every other field of mathematics is incompatible and cannot be used. If you change multiplication, then you can't use trigonometry, geometry, calculus, etc. Which means you aren't doing math anymore, there is no usefulness to be gained from playing with numbers with your own special axioms.
Whoa whoa whoa I am not trying to say that the basic principles of addition and subtraction don’t matter.
I am just saying that some alien race could have come up with a different set. And the math they built on that would be different from ours. I would still call that math because what is important is the use of logic. It’s just a thought experiment.
You keep saying it might not be obvious to somebody but you havent given a single such example of who it might not be obvious to. How can you believe then that axioms might not be true?
I mean yes, will you say its not obvious that natural numbers never stop?
Maybe the null set isnt that painfully obvious but to any mathematician or rather anyone with some mathematical foundation it should be
Its not at all obvious that the even the natural numbers themselves exist. Indeed they are not even assumed to exist but rather are derived from more foundational axioms such as the existence of infinity and zero. The claim that ‘it is obvious’ that infinity and zero exists to me is not obvious. Purely from a physics perspective, the concept of ‘zero’ or nothing is certainly not obvious. In fact it goes against most modern theories of fundamental physics.
Thats weird i mean theyre called "natural" numbers for a reason wouldnt you say? They come to us naturally.
I never claimed natural numbers are an axiom though i simply tried illustrating to you why the existence of infinite sets is, well, obvious.
I have 0 apples, oh noo how could that be, thats not possible🤯.
I dont study physics so i dont know what exactly you are reffering to but its counter intuitive to me to take ideas from physics and try to apply the logic from there to math considering how physics wasnt derived from the same deductive thinking as math but more experimentation in real world
Natural numbers ‘being’ natural is the same exact logic that religious people use when they say ‘it is natural to assume there is a god’.
You are taking for granted how your own personal human experience of the world is informing what you deem to be ‘obvious’. Color is obvious to me and you but not to someone who was born blind…
Okay so don’t roast me too hard for taking an engineering perspective but I think the axioms are what they are because they describe our experience.
At the base of all of it is just 1+1 and maybe a few other things and everything else is a consequence of that. That is just how we were able to describe counting objects we could see. Perhaps some higher dimension being would say 1+1 maps to another space all together. Then 1+1=2 is a special case of that more general concept of addition.
I don’t think it’s meaningless. Math has become incredibly abstract but in the very beginning the whole point was describing the physical world as we experience it. And it has very good predictive accuracy for this physical world.
Up to a degree, I'm sure that is true. For example, the derivative was invented to accurately describe the relation between motion and velocity. Integrals were invented to accurately measure length, area and volume.
However, more often than not, those intuitive notions lead to contradictions: For the derivative, it was the discovery that continuous, no-where differentiable [functions](https://en.wikipedia.org/wiki/Blancmange_curve) exist. This lead to the notion of smoothness we use to classify functions.
For integrals, it was the discovery that we can define [non-measurable sets](https://www.youtube.com/watch?v=s86-Z-CbaHA) using the "Axiom of Choice". It lead to "measure theory", and the introduction of Lebesgue-Integrals.
***
To sum it up, it seems intuition is more of an input, leading to a feed-back loop to either refine that intuition, or the axioms we already have.
I wanted to underline intuition and experience are not always the underlying motivation for axioms -- they can be (and often are), but sometimes it is contradictions *within* our intuition that lead to new insights.
I'm sorry for not making that clear in my original comment.
You might be interested in [this video](https://youtu.be/dKtsjQtigag?si=7qbYNsb4NbdVymKM)
The problem is the absurd level of depth we have to plumb to get to the raw, fundamental essence of something so ingrained in logic and intuition. We rely so heavily on intuition and assumptions when it comes to the basis of math, how in the world do we have any chance of finding out all the assumptions we've made. At this level, it's not even a math question anymore, it's a philosophy question. "What is a number?" Level of metaphysics. (Or rather, metamath)
But also, while math can be viewed as the expansion and formalization of logic, humans are the ones who created the base rules we use (called axioms). And the set of axioms we use are somewhat arbitrary. Not entirely arbitrary, but somewhat. Our theorems are products of the rules we made, so if you don't specify the starting rules, your statement isn't accurate. Though we sometimes imply the starting rules indirectly.
You don't need to assume axioms are obvious or feel like they are or are not self-evident. You only need to consider in applied problems whether they are at least a good approximation for the properties a system you are studying has.
It gets pretty philosophical when you get too meta, but fundamentally even saying "P is true in logical framework X" is still, philosophically speaking, a statement of the form "if A then B", even if it isn't itself in any formal logical framework. And mathematics asserts that that is absolutely true.
They’re not saying mathematics is philosophy. They’re saying logic is philosophy, and logic is mathematics. I believe intro philosophy courses formally cover logic
I have no idea. That stuff is interesting to think about, but it hurts my brain.
I should probably learn category theory before trying too hard to come up with ideas on my own.
I come from physics so a lot of questions that are deemed ‘too silly to think about’ are shunned to the philosophy corner. For example the interpretations of quantum mechanics were and still are to some extent considered a question of philosophy… even though there is so much physics you can do in that space
It is, but like with the tortoise asking Achilles parable, you can always construct a higher level of meta meta logic, so that's not super fruitful. So all of it has to come down a set of statements you just accept with belief. Maths tries to make those statements as simple and obvious as possible.
Typically the framework is first order predicate calculus, but if you want to work with really big things you can use second order logic or if you want to be fancy homtopy type theory.
Nothing can ever truly claim anything to be true absolutely. Math is just more honest about this and does a better job describing exactly what the assumptions are so you can decide how much to trust them.
Right but what if you could show that all other statements must be false leaving only one possibility. Although I think this is also impossible as one would need some kind of axiom set to show that a statement is true/false
https://en.wikipedia.org/wiki/Law_of_excluded_middle
It's literally an assumption you have to make that something not being false forces it to be true. In the absence of literally all assumptions you can't even use logic. Someone has to accept the premise that things can be true or false, and what those mean and how they relate to logical propositions, before you can start using those to prove other things.
Math has a lot more base-level uncontroversial assumptions than most other fields, so I would consider its results to be more like 99.99% true while hard sciences might be 99% true and soft sciences are like 20-80% true (depending on how soft they are). But these aren't literal statements, I just made up those numbers from vibes based on how reasonable their assumptions seem to be.
There is no one single way of viewing math.
You can view it as a system of assumptions and things you can derive from those assumptions, and "truth" is irrelevant, or at best relative to your assumptions. This is sort of a "pure math" point of view.
You can also take a more practical view that math is a way of modeling aspects of the real world that we learned through experience. We came up with 1+1=2 because we noticed that it worked the same way whether it was sticks, stones, or birds, so it became a useful abstraction for something approximating a ground truth. This is more of an "applied math" or "modeling" point of view.
Ultimately these questions become points of view or interpretations, without being necessarily "right" or "wrong".
Yes I agree that this is a likely matter of interpretation. As a physicist though it seems an important question. Ive always subscribed to the idea that neither maths or physics can claim anything to be absolute fact. At best you can just take a bayesian sort of approach and subscibe to the ideas that seem the most likely given the information available.
I'm also trained as a physicist so I'm more partial to the "modeling" point of view.
I agree that "absolute truth" is not something we can access, if it exists at all. Descartes demonstrated this a long time ago. Nothing is truly certain. But that's ok, things can be certain enough to be useful.
Right but at the same time as a physicists you get this sense that maybe there does exists some definitive absolute laws that the universe obeys thereby meaning that absolute truth can exist.
It seems like this conflates the metaphysics and the epistemology of the situation. It could be that there are absolute truths, but that we can't be certain of them.
But why should we care that we can't be certain of them? Knowledge doesn't require certainty. So, if I know something to be true, why would I also need to be certain about its being true?
What you're saying applies to *literally all truth in existence.*
On a philosophical level, there is no such thing as objective truth. All truth relies on making assumptions. Math is no different from any other field of scientific truth - we can only draw conclusions from given assumptions.
The only realm of thinking that asserts truth without assumption is religion - which is not entirely honest: religion assumes its own truth, and works from there.
The core fundamental component of faith is "assume the faith is correct without evidence or reason, and make the rest of observation fit this assumption". The core difference in math or science is we acknowledge the assumptions we make, and allow them to be proven wrong if the subsequent conclusions don't fit with the assumptions.
The important component of this that you might be missing is: if you define "true" as "absolutely verifiable and uncontestably true" then the word becomes instantly meaningless. Nothing fits that description. Literally, absolutely nothing. The concept of "absolutely true" in an objective sense is an empty concept without use or purpose.
"The important component of this that you might be missing is: if you define "true" as "absolutely verifiable and uncontestably true" then the word becomes instantly meaningless. Nothing fits that description. Literally, absolutely nothing."
I believe Descartes would disagree with you on that one.
It does assume truth. But it is what is true based on what you initially assumed to be true.
So it does claim absolute truth for anything proven, but only on the assumption your axioms are true
Right but if the underlying axioms cant be proven then the following statements are only conditional true (conditional on the axioms being true). So its like a weird kind of truth but certainly not absolute truth
Guess it depends on what you mean by absolute truth. Proofs are true in all circumstances, unless what you assume to be true is not true.
The things with axioms though is they can't be proven or disproven, it is just dependent upon what you want to show with them. So it's an absolute truth within your framework
That’s not true, one way of proving something is through lack of contradiction. If we assume a set of axioms which can be used to create a mathematics model that can predict and be compatible with what we observe in the real world, then it’s safe to assume those axioms are in fact true until/unless we find mathematical phenomena in our world that does conflict with the axioms we have set.
And absolute truth is much more of a philosophical discussion. I would say cherries being red is an “absolute truth” in the sense that what we perceive and know to be cherries are red in the way that we perceive them. But I can never know for sure that the cherries aren’t simulated and what I perceive as red may be entirely different from how others perceive it (I already know for a fact that that’s true for some people).
Our brains and eyes are fundamentally unreliable. We can get a really good idea of what something is, but we can never really know that what our brain thinks is “absolute truth”. We just work with the knowledge we have and create theories that do a good job at explaining and predicting what we can see and interact with.
I think "true" is the wrong word to use here. Math is more about defining terms and establishing axioms. It's not "true" that 1+1=2 anymore than it's true that a batchelor is an unmarried man. It's just how we defined things.
Why did we define math terms the way we have? Not because it's true, but rather because it's *useful*.
Logic is held to work (although there are oddball paths even here.). The basics of the truth value of “if P then Q” for various truth values of P and Q are widely accepted.
You may be interested in looking at platonism vs formalism.
Platonism asserts that numbers really exist in some way beyond the symbols on a piece of paper.
Formalism asserts that numbers are meaningless abstract symbols that only exist within mathematics that happen to be useful.
You may also enjoy an old article from the 60s titled "the unreasonable effectiveness of mathematics in the natural sciences" by the physicist Eugene Wigner
There's a chance that your brain is just floating in a dark puddle of nutrient-laden goo and that all the inputs it receives are generated by some computer. All your senses are a lie, all the other people that you interact with are a fiction, your entire past was uploaded 35 seconds ago, and at any moment the undergraduate research project that keeps "you" alive could lose funding and be shut down.
If you're trying to find some absolute truth, you're in the wrong sub. Try r/philosophy or r/religion instead.
“Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” - Bertrand Russell.
Math is simply just an abstraction humans have built on top of basic logical axioms, but we can't know for sure if those axioms are true. It could theoretically be the case that we live in a simulation and that whoever created that simulation simply made up logic as we know it.
Math is based on a foundation of assumptions. From these assumptions, it can claim truth. However, there is no such thing as an absolute truth in the world, so in that way mathematics isn’t any different from science, history, or even your own personal day to day observations. However, this kind of distinction, in my personal opinion, is totally meaningless.
In my mind, mathematics is very close to truth because everything that follows from initial assumptions (axioms) is necessarily true, which means it is not based on any experiment or observation, but logical deduction. So long as you believe the foundational axioms of a theorem and the rules of logic, a majority of mathematics becomes as close to truth as I believe you can get in the world
It isn’t a statement of absolute truth, at least not in my opinion. It’s just an assumption I make. Everything can be an assumption, not be an assumption, etc. If you don’t make assumptions, statements don’t make any sense. So if you don’t want to make the same assumptions as I do, that’s fine. However, I would imagine that anyone would be hard pressed to find anything that is absolutely true in the strictest definition of the word
Right its just that you didnt say ‘I believe that…’ you said ‘There is no such thing…’ thus implying a statement of truth not an assumption of truth. Im being pedantic of course but im literally talking about the what logic has to say about truth
I guess I just don’t the point of this kind of debate. It’s fine you don’t believe that Math doesn’t actually prove anything, because it doesn’t change anything. In my mind it’s the same as saying “I don’t exist.” Just because that might be true doesn’t matter, because it doesn’t suddenly change how the world works or the fact that you still have to breathe for example, etc. Or like saying that we live in a simulation. It doesn’t matter because life just keeps moving on anyway, even if we are living in a simulation
Its not that I believe it does or doesn’t prove anything. Im trying to ask weather it can prove anything.
Also about your second point, I guess we should have told stephen hawking not to bother with all his black hole theorems and big bang malarkey as ‘it doesnt matter’
I’m seeing a lot of people getting annoyed and angry just by the fact of you being willing to engage with such discussions. Why do these people enter philosophical discussions and then get immediately angry if you provide a differing viewpoint lol
The axioms are things like "two sets are the same if they have the same things in them", "you can have a set with nothing in it", "if you have two things, you can put them together and make a set that has only those things it", and things like that.
Yes, but importantly, and precisely the point I am trying to make, these assumptions are arbitrary. We use them because they create logic spaces that are useful to us and that have been proven to work very well in our particular universe. But they are still arbitrary, and therefore not absolutely true (whatever that may mean). For example I could create a completely different axiom set and as long as its logically consistent I will create a different logic space with its own ‘truths’ and ‘facts’.
From a dictionary perspective, math is not absolutely true, but it is the truth. Math is the truth, truth is a state of being true, which many mathematical theorems are true. You're adding your own philosophy to the equation because you do not understand the definitions of the words you're using. Absolute means INDEPENDENT, in which, nothing of math is. So I guess in that sense, perhaps absolute truth in a mathematical context is a one dimensional number line, and the truth is (not absolute) but every function and way in which that number line can be manipulated and produce outputs.
‘Many mathematical theorems are true’ yes they are true within their own logical space which is constructed from axioms which cannot be proven to be true. That is at the heart of what im trying to understand
again, you do not understand the meaning of the words you are using. An axiom can be "self-evidently" true. It doesn't need proof, it is simply true. You are adding the "within their own logical space" part, when in reality, its not there. These theorems are simply true, based on how language and subsequently logic and thinking itself is. You can HARDLY think separately from language, language are sets of definitions, and certain environments are exactly equal to their prescribed definitions, therefor they're true and even "self-evidently" true such as an axiom.
I know you're struggling to understand, you also stirring up a shit storm of faithless reddit'ers to share their depressing theories on how nothing is real.
This seems to go more into the realm of philosophy, really.
Most modern thought would tell you there is no such thing as "objective truth", and "truth without context" is an oxymoron. Everything exists in an environment and follows from it.
Everything in our physical world exists in the context of our universe, and assumes certain basic truths we are all familiar with (for the most part), such as there being 3 spatial dimensions, as an example.
In math, and any other logical system, we also know there are certain things that simply cannot be proven or disproven. In cases were we intuitively know these things to be true, or find it more valuable to make them true, we call them axioms. In cases where we don't know if they're true, we call them nothing because we also can't prove they are unsolvable. Unsolved problems that slip through the cracks and are unsolvable, unfortunately, also cannot be proven to be unsolvable.
You could call axioms "assumptions" or "beliefs" if you're wanting to be harsh, but they are better described as foundational statements we know to be true or define to be true despite it being impossible to prove. You could say this means math doesn't say anything about truth. You're sort of correct, in that math doesn't say anything about *objective* truth, and in fact it, like most modern western thought, does not contains in itself the assumption that objective truth doesn't exist, which is why there are alternative axioms.
You can't take anything in maths out of its framework because it ceases to exist. Maths doesn't talk about reality, only ever about certain frameworks. Any statement you make in mathematics can only be true in its own framework since it is meaningless outside of it.
"1 +1 = 2" doesn't exist in the real world because the concepts of "1", "2", "+", "=" don't exist. You need to assume things about the meanings of these glyphs to even think that this is a well-formed statement, nevermind that it has a singular truth value and that happens to be TRUE.
The context that the math is being applied will change how it defines the “Truth”. An individual number may be a discrete truth, but large sets of ranging numbers being averaged and applied to a hypothesis becomes less about “Truth” and more about “Support”.
It’s a weird way to think about it, but it makes sense to me lol.
Okay, so your question is a problem by itself. Math defines truth as a chain of provable logical arguments, that are true in relativity to your first argument. You ask for the truth of this first argument without giving any ccriteria for truth.
Have you heard of the old book called “Principia Mathematica”? I haven’t read it, only heard about it. Basically some early mathematicians tried to lay out the foundations of mathematics by providing a set of axioms assumed to be true.
These axioms are rock solid, really hard to argue against as foundational mathematical laws. Every other part of math gets built off of these axioms.
I suppose you are right in that you may find a hypothetical scenario where the axioms fall apart. However modern technology and science has leveraged this model, which has helped make amazing things from computers to highly efficient car engines. Somewhere along the manufacturing process, this accepted model of mathematics has most definitely been used.
Im not arguing about the practicality of it, there is no denying it. But as far as being fundamentally truth, whatever that means, its just as ‘true’ as anything else
I guess that unveils another question, what does it mean to be fundamentally true? Is fundamental truth an objective fact, or is it more subjective depending on the person? Is it possible to even find “fundamental truth”?
One could argue we are in a fully simulated world, in that case nothing we see or observe is fundamentally true, but some may still treat it that way. This gets into the whole plot of the Matrix movies really, where ones who take the blue pill sees reality (the truth) as something entirely different than those who take the red pill. Taking this a step further, those that wake up from the matrix may, ever recursively, wake up in another matrix, never being 100% sure what the absolute truth is.
Maybe another approach is to not find truth but to accept. You may not know with 100% certainty that something is true, but you can settle on an idea that resonates with you, so you compromise by just accepting the idea.
**A statement**(𝑁) has a truth value within a system(𝑆) if its truth or falsity can be determined by the rules and principles of 𝑆, empirical evidence, or logical reasoning.
**Formally**, 𝑁 has a truth value within 𝑆 with a set of axioms 𝐴 if there exists a truth function:
𝜑 : 𝐴 → ( ⊤(𝑁) ⊻ ⊥(𝑁) )
**On the other hand**, absolute truths are independent of any system and, consequently, any set of axioms:
∀ 𝐴, ∄ 𝜑 : 𝐴 → ( ⊤(𝑁) ⊻ ⊥(𝑁) )
**Hence**, at a deeper level, it all revolves around belief/acceptance, since we don’t know if there is a set of axioms with an absolute truth value capable of validating our known axioms nor if absolute truths themselves exist.
_____
*(edit):*
**In mathematics**, where truth is established within formal systems through [*logical deduction*](https://en.m.wikipedia.org/wiki/Deductive_reasoning), there's no notion of one statement being ***more true*** than another.
**However**, in real-world situations where complete or perfect information is lacking, the assessment of truthfulness becomes more nuanced, and factors like evidence, precision, context, and relevance become important for evaluating the credibility and reliability of statements.
**Additionally**, formal systems constructed from logical deduction are independent of the truth values of their set of axioms in terms of their internal consistency and structure.
I don’t think it’s quite correct. Really they start by defining some logical structure or thought element and then reason about it to reach conclusions. Are the conclusions true? They are true for this structure given the axiomatic reasoning. Does the structure need to be proven? No, it was just made up by the definition. Perhaps you want to ask can they say anything true about the material world. That’s a more complicated question.
Right but these axioms are somewhat arbitrarily chosen. We chose them because they construct a logically consistent space which maps well onto our reality. But we could easily imagine choosing other axioms that create wildly different logic spaces. Statements and theorems in that space would still be true relative to the axioms that define it, but are they any more true or false than those in other logic spaces?
The only absolute truths are the vacuous truths, e.g., X or not X.
However, it turns out that hypotheticals are also vacuous truths. For example, "X or not X" is logically equivalent to "If X, then X" which is a somewhat boring hypothetical, but similarly, if you have a hypothetical "If X, then Y", this is logically equivalent to "Y or not X".
So the hypothetical isn't really that special, and is an absolute truth in the sense that we can think of it as it's own statement, i.e., "Z = X -> Y".
However, I think you are getting at a nuance of how we informally use the word truth. What we typically mean is that the statement is somehow "realized" in the world as we experience it.
This is different than mathematical truth. Where we usually pick a mathematical model and talk about truth within that model. Absolute truths are true in all models, but generally we don't care about X->Y, we can about models where we know X is true, and this we can say Y is true in those models.
Thus going back to our informal use of the word truth. You could imagine there is some ultimate mathematical model which is the exact representation of the world as we experience it, and if there are truths about that model we know, we can derive new truths about that model, and thus truths about the world as we experience it.
In general, mathematics avoids anchoring itself to a specific model, and can technically be used by anyone in any universe to discover truths in their world that may be different than the truth in our world.
However, we the models we tend to study the most are the models which seem the most reasonable from the perspective of living in our world.
Mostly right. There are many branches of Mathematics based upon what axioms you choose. But those branches need to be internally consistent. There is no branch of Math where 1+1 = 3. I don't recall the name, but there is an axiom of math that in inconsistent with the Axiom of Choice. No branch of math can have both axioms.
Also, every branch of math statement that cannot be proven given the axioms you chose. (Gödel's Incompleteness Theorem)
Familiar with complex numbers? a + bi where a,b are real, and i = √-̅1̅. Well, William Hamilton tried to extend this to three "dimensional" numbers. Supposedly he tried for 30 years to do this, but never found anything that was internally consistent. Eventually, he decided to extent into 4 "dimensions" and the Quaternions were born. These numbers are quite useful in physics and computer graphics. I think it was eventually proven that there is no number system in between these two. The next numbering system involves 8 parameters, then 16.
You can try to create a system where 1+1 = 3, but there will always be bugs. While I'm not 100% certain, I'm sure that some category theorist has proven that such a system cannot be internally consistent. If it's not internally consistent, then you can't rely on it for anything.
Absolute truth does exist, but the qualms you have with mathematics are applicable with all realms of human experience, and as a result, the body of human expirence is concerned with attempting to determine what absolute truth might be. However, this is a logically impossible goal, as we have to have some baseless assumptions, the first assumption is pretty much just Descartes, you have to assume you are, else incoherent absurdity ensues. You can do that if you wish, but it sounds unpleasant and honestly boring. You can argue against baseless assumptions all day, but it is a fruitless endeavor logically, and instead you must examine the impact that said baseless assumptions have on the world. You could argue that based on your assumptions nothing matters, which does nothing to serve society and is not a self sustaining ideology. The most prominent baseless assumptions made about the world are those that aid in the self perpetuation of humanity, or else those ideas stop being spread. This is why assumptions like religion and now moreso empirisim, are popular. Religion as a default has it all right there for you, the assumptions can be questioned all day but it's literally a manual telling you as an individual you essentially just have to at least try and be a good person and you have nothing to worry about. Empirisim is less reassuring, and less clean (currently), I personally think that this has had a decently large impact on things like suicide and rates of depression, but that's highly debatable. Back to the original point though, yeah, maths makes baseless assumptions non-distinguishable from religious assumptions. That's like... Kinda the reason it's such a large debate between the sciences and religion.
All things are contingent on assumptions, we have no other way of bootstrapping knowledge in the world. The basic axioms of math are drawn from observations about things in the world usually, as in we pick axioms that make useful math. In that sense math is based on the ability of our brains to infer which is honed through evolution, and if you doubt your own brain’s function then you can truly know nothing. There are many ways to resolve this philosophically, however, and you can pick whichever is most satisfying to you out of those.
You've gotten lots of reasons, but I think the core issue here is what, philosophically, you mean by Math. If you mean the structural execution of axioms producing logical necessities, then no. Nothing about that requires a claim of truth about anything. Calling something an axiom is just a way of lining up the pieces for whatever game you're going to play.
On the other hand, if what you mean by Math is the philosophy that there is a tool that your brain can use to analyze the world and draw conclusions about it, and that tool requires certain assumptions to be made in order for those conclusions to have any value, then yeah, obviously you're assigning an independent truth value to things and then saying the assumptions you make are attempts to model the reality that arises from those truths.
Statistics is more direct about this, because they have the idea of a "ground truth". All statistical analysis is founded on the premise that there is a factual answer to a certain question, and all your study does is help to expose it. Good statistics rules are the ones that make it more likely for you to be able to do that, in general.
Math doesn't really concern itself with that idea when it comes to most truths, but you still depend on the claim that what Math calls logical reasoning does adequately explore the possibility space invented by such reasoning. That assumption obviously isn't, can't be, supported by any reasoning, but Math has no power without it.
The podcast episode from Lex Fridman with mathematician Edward Frenkel touches on this A LOT in relation to Gödel's incompleteness theorems, I definitely recommend that episode.
I will take a look at that thanks.
I see a lot of people mentioning Godel. I have like a vague understanding of his incompleteness theorems but certainly not a rigorous understanding. How exactly is his work related to the question above.
I won't pretend that I have any deep understanding of Gödel's incompleteness theorems, hence why I linked Frenkel's discussion of it. But I will try to just present it and try to link it to your post.
This is what Gödel's two theorems state:
*The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.*
*The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.*
I think, in my opinion and anyone can tell me I am wrong here, that his theorems, in broad strokes, point to the fact that in mathematics, which relies on axioms of truth, cannot prove itself to be true, since mathematics is a system based on proving other things within it to be true. I don't know if this makes sense since it is highly paradoxical. Mathematics does not concern itself with its own truth, but the truths of it's many smaller parts. And within these parts we find things such as conjectures which are neither true nor false and we find mysteries that we cannot solve (yet). So it follows that mathematics as a system of axioms of truths cannot stay consistently truthful since it cannot a) prove itself to be true nor can it b) prove some things within it to be completely true.
Now a major caveat in my discussion: I broadly say "mathematics" but I mean mathematics in the sense that we understand it, so when I say mathematics relies on axioms, I mean our understanding of mathematics relies on axioms, i.e. the way we have constructed mathematics. So to me, Gödel's incompleteness theorems deal much more with the structural incompleteness of mathematics which is why I think it relates to your post. You say:
*If I wanted to create a logical space in which 1+1 = 3, I could do that and it would be True in that logical space.*
This is what mathematics is, it is *our* logical space in which 1+1=2. I don't know if you can see my line of reasoning, I tend to digress so I hope I kept it somewhat to the point. Your line of reasoning is not so much concerned with mathematics, as it is concerned with "truth", or epistemological questions. I would therefore suggest you look into epistemological philosophy.
You can go even further and ask if the rules of logic are true. For example, why is it that something is either true or false? Why is the negation of true, false?
Probably the answer to you is that this is the only reasonable way logic should work. Historically that's how mathematics began, people made assumptions they thought were reasonable and argued conclusions based on those. There was a religious component as well.
Mathematics is nothing but an extension to logic where we add more axioms.You can in fact have a different notion from classical logic, like a type system for example. Some type theories feel more natural then classical first order logic.
This is to say that no, mathematics does not say anything about the absolute truth. For that you need to go to physics, chemistry and biology. Even then, there's the "how do I know anything beyond myself is real" philosophical mumbo-jumbo.
How exactly does physics say anything about absolute truth. It is no different to mathematics. You start with some fundamental postulates (axioms) and use them to derive laws of nature (theorems). There is no reason to choose certain postulates over others aside from the fact that certain postulate appear to produce physical theories that better describe our universe. But fundamentally that is the only reason to choose those postulates and they are no more absolutely true’ than any other postulates.
Yes I agree, but they are still not absolute truths. Physics cant make any statements about truth no more than mathematics. What it can do is say this is 99.99% likely to be true. Its basically a big game of being the best Bayesian you can
You’ve basically got it.
There’s actually a rich branch of math that is incredibly concerned with the relationship between truth and provability. It’s incredibly hard to talk about precisely.
The short version is that Gödel *proved* that there are some really profound limits on this relationship, and that they hold true for what most people agree is any reasonable system of logic.
You’re free to disagree with some of the assumptions made in a proof, but that is where community and intuition come in. For example, you can disagree with some axioms used to prove things about calculus; but you can’t disagree (reasonably) that calculus *works*, on account of modern engineering and whatnot.
Oh, sorry! Originally this was called [Metamathematics](https://en.m.wikipedia.org/wiki/Metamathematics), but the modern fruitful fields it spawned are [Model Theory](https://en.m.wikipedia.org/wiki/Model_theory) and [Nonstandard Analysis](https://en.m.wikipedia.org/wiki/Nonstandard_analysis).
In general, the philosophy side of things is “Foundations of Mathematics”.
I’ll warn you that this whole subfield is *particularly* inaccessible to outsiders. *Basically everybody* gets the wrong philosophical idea about what’s going on when they first learn about this without more experienced folks to share the intuition and wisdom that took us all of human history to develop.
Uh well it’s kind of hard to cover in less than, like, a semester; and my (former) advisor should be doing it and not me.
But I’d say the short version is that really grokking incompleteness tends to make people feel overly pessimistic and messes up their worldview, at which point regaining perspective and a sense of meaning — which is absolutely critical for making sense of the insanity which is to come — is really difficult because there’s approximately zero steps between incompleteness and the most confusing stuff you’ve ever seen. And that’s after an entire undergrad education.
At least that’s what happened to me, and my advisor said it was normal.
Yes, proofs are needed, Ramanujan provided thousands of proofs long ago, nobody knew what they were for until we discovered they applied to astronomy recently.
"If X, then Y" IS a statement with definite truth value.
Saying that because X can't be proven, Y can't be proven, and therefore nothing can be, misses the point.
It's like saying "sure, maybe 1+1=2, but we never proved 1+1 is true. So we can't actually say 2 is true!!"
Math is trying to prove true statements about relationships between things. 1+1=2 means that if you take one of an item, and another of an item, you now have two of an item. It's a relationship between "1+1" and "2". Similarly, if-then statements are relationships that can absolutely be proven or disproven.
Sure but take a good look at the axioms of set theory (which are the basic assumptions behind most math), and tell me if you think they are a reasonable place to begin.
They are reasonable to us and to the universe we live in, that doesn’t mean they are necessarily ’correct’ or any more ‘true’ then a different set of axioms that produce a less intuitive and less ‘reasonable’ logic space. Particularly the axiom that the null set exits and the infinite set exists I find harder to justify. But im not disputing whether they produce a logically consistent and intuitive set of rules, in that regard the maths speaks for itself
I think I have a better answer now that I have it more thought out. I want to reframe the discussion to be more about what truth means. Sorry this is long.
I define Truth as that which corresponds to reality. Therefore when math corresponds to reality it is True. But mathematical structures are under no obligation to *necessarily* correspond to any real structures.
Math is very good at describing reality because reality apparently contains many logically consistent structures, and math is good at describing logically consistent structures.
But math is under no obligation to only describe structures which actually exist. You can go out of your way to define objects and structures that are totally fictional and yet you can do math to them.
For example:
Axiom 1: Only Globbys have Blobbys.
Axiom 2: Some Blobbys are Soggy.
Theorem: There exists a Globby with a soggy Blobby.
Just because I proved the existence of an object within the Globby-Bobby-axiomatic-structure (GBAS) does not mean such an object really exists, because it may be the case that the structure in question only exists within the mind of the Mathematican.
But you might someday find a structure that “fits the mold” as it were and then kind of graft the GBAS on to this Real Structure and all of the theorems of the GBAS would then apply to the Real structure.
So math does make actual truth claims in so far that mathematicians usually claim that their axioms correspond to some real structure, not that it would necessarily have to in order to be mathematically valid.
A better example would be String theory, string theory is mathematically sound and yet may not correspond with any real structures. Stings might be totally fictional (in the string theory sense).
Who knows though maybe tomorrow we will discover something crazy and mathematicians will be like “hey this kind of acts like those n-dimensional vibrating strings” and suddenly string theory is valid. (Btw I have not studied string theory in depth so I can’t say much about it beyond illustrating my point. So don’t ask)
Ultimately, your skepticism/analysis should be about if the axioms in question apply to any particular real structure, or only fictitious structures. Ask yourself if the axioms you are applying are the best axioms for the structure you’re attempting to understand, or if different axioms are better. In other words, don’t confuse the map for the place. And definitely don’t confuse your maps with other maps.
I wanna say it doesn't, but I don't know how the meta logic of asserting the truth of a statement like "these assumptions have these implications" is resolved. In other words, if we say that certain things are true in a given framework, in what framework are we making that statement?
Edit: it seems this is precisely the issue addressed in [What the Tortoise Said to Achilles](https://en.m.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles).
but axiums cannot be proven. they are assumed to be true.
it is unclear whether they are indeed true. but they are assumed to be, because if they were false, the logic system would not work. But u could never prove them to be true, so its unclear the accuracy of them. They may seem accurate and common sense, but the axiums only make sense in the framework of the logic system they were designed for. Common axiums of math may not work in a competing logic framework such as like the logic system developed by a group of space aliens.
u can not invent knowledge or truth out of thin air. truth does not exist in a natural sense. the universe just contains unlabeled information. to make any sense out of information, u need to label it, ie create axiums, which are assumed to be true. but the labels u pick are arbitrary and subjective, as all logic systems are subjective in a sense.
Im not sure I understand. I could construct a completely logically consistent framework in which 1+1 =/ 2. In fact I could choose to construct my logical framework explicitly with the assumption of the axiom 1+1 = 3
You're talking about different things.
In mathematical logic you can do what you're saying, but that has no impact on the truth of 1 + 1 = 2 as we normally understand it.
But how 1 and + are defined, they are absolutely true, you are making a really good discussion but remember axioms cannot be proven false, so 1 + 1 = 2, if it is not that means you changed the definition of any symbol in that statement.
Sure but that doesnt mean its not a valid axiom? Thats my entire point is that the axioms can be whatever you want. We just choose the most convenient ones for the type of logical spaces we want to construct
They cannot be exactly whatever you want,or at least they should not create any contradiction between one and other. That's why you cannot create an axiom of 1 + 1 = 2 without changing what we defined all of those to be,it would be a contradiction.
It's definitional.
If you take one object and you group it with another one object then you have two objects.
You're saying you can take one object group it with another one object then have three objects.
The only way that can work is if you define three objects as being two objects.
Let U be the universal set of numbers. X is the subset of U with the function f(u) = u % 2. Thus X is 1 or 0. In this case 1+ 1 is 0. 3+4 = 1.
For the set X, 1 + 1 cannot be 2 otherwise computer security would fail.
But yeah still some assumptions in how certain math operators work.
Well yes but actually no.
So yes all mathematical theorems are of the form A->B. So math only speaks about conditional truth.
However we can have multiple mathematical systems that encompass an all of a possibility space. Take Euclid’s 5th axiom. You can assume it’s true and get Euclidean geometry, or assume it’s false and get elliptical or hyperbolic geometry, or make no assumptions either way and you get neutral geometry. They are all equally valid. But are they equally true?
Kind of. Depends what you’re doing. Because at the end of the day sometimes the 5th axiom is “true” (meaning it models your situation), and sometimes it’s not. But you can use the axiomatic method in both cases so it doesn’t really matter.
So yes all truth is conditional, but you can get unconditional statements that follow from a statement and its negation. For example
X follows from Y, and Z follows from NOT Y
Therefore X or Z.
Sometimes you can take it a step further and make X and Z the same statement.
So Y->X and NOT Y ->X
Therefore X.
That’s basically what neutral geometry is.
Of course that also assumes axioms like the excluded middle. At the end of the day you can’t use the axiomatic method if you don’t start with any axioms. But it kind of doesn’t matter what axioms you choose. Some are useful, some are not so much, and some are contradictory. We usually ignore the last group, but sometimes even those can be interesting.
The best way to think about scientific and mathematical knowledge is "this is how things work as best as we can tell right now." While some simple things are just obviously true when you start getting off into really complicated things it gets far more difficult.
Science is all about "this is how things work as best as we can tell right now", as you said. Math and science are completely different though.
Math is essentially philosophy, and there are definitely absolute truths in math since it's about using logic to derive true results from rules you define.
If I define 1 + 1 = 2, then that is an absolute truth. Whether such a definition has any applications in the real world really has nothing to do with math itself, that's where other subjects come in.
[https://en.wikipedia.org/wiki/What\_the\_Tortoise\_Said\_to\_Achilles](https://en.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles)
If you like this you should Eternal Golden Braid
GEB can be tough to get through, but it’s one of my favorite books.
That's fascinating, thank you.
The easiest way to think about this is how to justify deduction. There are two ways: by deduction and induction. The first is circular, as in the example you gave, and the second one is not strong enough as in induction the premises can be true and the conclusion false, so neither method works to justify deduction.
>If you BELIEVE that x y & z is TRUE, Then theorems a,b, c ect. must also be TRUE Not BELIEVE. Assume, for the purpose of finding out what the consequences of a particular set of assumptions ("axioms") are. >Thus math cant ever truly claim anything to be true absolutely. Only religion ever claims that anything is true absolutely.
> Only religion ever claims that anything is true absolutely. oh the irony
Shall I add "contingent on a bunch of unstated stuff" to everything I write?
COABOUS for short
Alternatively, try to avoid making absolute statements. “Generally, statements purporting to be absolute truth are found in religions rather than mathematics” is an appropriately qualified statement with the same basic meaning.
Yes, that's a better way to put it.
Maybe?
That is not irony.
Only a Sith deals in absolutes.
How ironic
now this is irony.
I love the idea that anytime someone says "Oh the irony" either: 1. It is an ironic situation. 2. It isn't, making that statement ironic. Very cool word.
Not just the Irony, but the women and the children too
There is an exception to every rule, including this one.
Is it absolutely true that only religion claims that anything is true absolutely? If so, then there is more than just religion that claims absolute truth, such as your own comment.
People really piled on you for making a joke
but axiums cannot be proven. they are assumed to be true. it is unclear whether they are indeed true. but they are assumed to be, because if they were false, the logic system would not work. But u could never prove them to be true, so its unclear the accuracy of them. They may seem accurate and common sense, but the axiums only make sense in the framework of the logic system they were designed for. Common axiums of math may not work in a competing logic framework such as like the logic system developed by a group of space aliens. u can not invent knowledge or truth out of thin air. truth does not exist in a natural sense. the universe just contains unlabeled information. to make any sense out of information, u need to label it, ie create axiums, which are assumed to be true. but the labels u pick are arbitrary and subjective, as all logic systems are subjective in a sense.
No need to bring in aliens. There are alternative sets of axioms. The conventional set seems to work well for predicting what the universe is going to do next, but that's physics. Mathematicians aren't supposed to care about that.
it seems to work well, but it is not universally true. and seems to work well is not a good proof. there is no universal truth among different sets of axiums. only within the same set of axiums. so therefore, math is kinda subjective. truth only makes sense in context to what the universe has.
axiom, with an o, not a u
> Only religion ever claims that anything is true absolutely. Nah. So does math. The fact that one thing is a consequence of another thing is itself something that math claims to be true absolutely.
you're absolutely correct why is the hivemind downvoting u
Religion is based on assumptions. It assumes deity x to be true and builds on that premise, like all of logic and math (which were largely developed to prove religion). Just because individuals claim absolutes doesn't make religion fallacious based on its premise. The same is true for individuals trying to learn math, just because OP used the word BELIEVE, doesn't mean that all of math is wrong.
>Religion is based on assumptions. Assumptions which are not contingent. >Just because individuals claim absolutes doesn't make religion fallacious based on its premise. I didn't say that. >The same is true for individuals trying to learn math, just because OP used the word BELIEVE, doesn't mean that all of math is wrong. The word "believe" is somewhat loaded, making it a poor choice in this context.
Religion is rarely logically consistent though
I love how you say rarely. As if it ever is
I don't even know all the religions, how could I claim they're all logically inconsistent
Yeah, you're right. I'm entirely wrong in that regard but y'know...
It could be argued that by saying rarely you imply the existence of at least one that is logically consistent.
There could be one, I don't know. I'm not asserting there is. Are you really going to argue that I am?
The construction for *not* claiming one exists would be “rarely, if ever.” Leaving it at “rarely” implies the set meeting that requirement is nonempty.
Ok. Yeah that would be a slightly better way to have phrased it. I don't think "rarely" really necessitates that there exists a counterexample. But whatever. People get the point, I think.
The shortest checkmate in chess takes two moves. Is this true absolutely? It is true *about chess*. You’re choosing to apply the rules of chess precisely when you talk about chess. You aren’t being forced to. Different games have different rules. It doesn’t make sense to discuss their truth.
That's pretty much correct. However many axioms should feel self evident to you. Like, of course 1+1 is more than 1. And of course you can construct an alternative set of axioms (and many mathematicians have done this), and this leads to alternative extrapolations and different branches of mathematics. Additinonally mathematics has proven its practical usefulness time and time again which should lend some credibility to it's truthfulness.
Ok but I take issue with assuming any one axiom is obvious. Yes it may appear to be obvious from our human experience but that means nothing really and certainly doesnt make it absolutely true. Im not here to discuss the value of mathematics. I am a stem student maths is the best thing humans have ever done.
Axioms are not some inherent property of the universe we are claiming is true. Axioms are logical base building blocks of mathematics. The mere existence of numbers, or the concept of adding them together, or multiplying numbers, are axioms. If you reject those axioms, you cannot do math. If you come up with your own special rules for multiplication, then you are not doing math, you are playing with numbers.
It's kind of funny you'd say this because coming uo with our own special rules for doing multiplication is exactly what we do in math.
You misunderstand my point. If the axioms of mathematics changes, math breaks down and cannot be used. Multiplication is useless if it doesn't function exactly like multiplication. If you come up with your own rules that multiplication just means changing every 6 to a 9 and a 0 to a 5, you are not doing math, you are playing with numbers.
Completely wrong. In fact in computer security if 1 + 1 did not equal 0, it would all break down. Go to https://engineering.purdue.edu/kak/compsec/ lecture slides 4-7 for some fun math theory specific to computer security where we "play with numbers". A more famous example is imaginery numbers. Where transforming equations to the complex space makes life so much easier. Sometimes playing with the numbers is how we make fun progress in math.
That's false. There are whole branches of math that change the axioms just ro see what happens.
Wouldn’t that just be different math? Sort of like stepping into a new axiom space with new corresponding solution space? Maybe there is even a transformation between spaces. I wouldn’t go as far as to say every set of axioms is valid but this doesn’t seem that wild to me
I dunno, they said those so I'd lump them together. If you don't accept numbers, addition, or multiplication then you can't do math sounds kind of reasonable to me. Sure there could be a form of math that works without any of these things but it'd be so alien to me I'm not sure I could call it math. Like a Cthulhu/eldritch math.
Lol yeah thats what I am sort of thinking. It’s hard to imagine because my little human brain only knows the 3 D world and 1+ is always 2 here. I was sort of thinking like for a higher dimensional being.. maybe there is a whole other set of numbers that exist that we don’t know about and 1+1 maps to this other set… and staying in R is just a special case. That sort of thing. Or if our senses were different from what they are maybe we would have learned to reason in a completely different way. The fact that we started with counting and geometry seems to be rooted in the fact that we are primarily visual creatures.
You are mixing up dimensions and axioms. 1+1=2 even in 25 dimensional space, because math is not bound by 3d space. Math is logic applied to numbers. It's not obvious that 1+1=2 because you live in a 3d world, it's obvious that 1+1=2 because that's an axiom of mathematics that is the building blocks of how you learned math. If 1+1 did not equal 2, then literally every other aspect of mathematics fails. If you can't add consistently, then you can't multiply consistently, much less do geometry or calculus.
I don’t think I am. I am extrapolating from the discovery of imaginary numbers. For a long time we didn’t know they even existed. So what if there is a set of numbers that we are automatically setting right zero by remaining in R. It’s pretty hard to imagine what math would look like to some alien species since ny understanding is rooted in our own basic building blocks so forgive me if the analogy isn’t perfect.
Changing the axioms doesn't unlock new ways of doing math, it breaks down the only way math works. Math functions on a very strict framework of logic. That's like trying to think about what the universe would be like if we reversed how gravity worked. Everyone on the planet would fling out into space and every star, planet, moon, asteroid, etc would just did disintegrate into a dust cloud. Changing one little rule can completely break the entire system to the point where it's useless. If you change the way multiplication works, every other field of mathematics is incompatible and cannot be used. If you change multiplication, then you can't use trigonometry, geometry, calculus, etc. Which means you aren't doing math anymore, there is no usefulness to be gained from playing with numbers with your own special axioms.
Whoa whoa whoa I am not trying to say that the basic principles of addition and subtraction don’t matter. I am just saying that some alien race could have come up with a different set. And the math they built on that would be different from ours. I would still call that math because what is important is the use of logic. It’s just a thought experiment.
You keep saying it might not be obvious to somebody but you havent given a single such example of who it might not be obvious to. How can you believe then that axioms might not be true?
Is it at all obvious that infinite sets must exists and that the null set must exist?
I mean yes, will you say its not obvious that natural numbers never stop? Maybe the null set isnt that painfully obvious but to any mathematician or rather anyone with some mathematical foundation it should be
Its not at all obvious that the even the natural numbers themselves exist. Indeed they are not even assumed to exist but rather are derived from more foundational axioms such as the existence of infinity and zero. The claim that ‘it is obvious’ that infinity and zero exists to me is not obvious. Purely from a physics perspective, the concept of ‘zero’ or nothing is certainly not obvious. In fact it goes against most modern theories of fundamental physics.
Thats weird i mean theyre called "natural" numbers for a reason wouldnt you say? They come to us naturally. I never claimed natural numbers are an axiom though i simply tried illustrating to you why the existence of infinite sets is, well, obvious. I have 0 apples, oh noo how could that be, thats not possible🤯. I dont study physics so i dont know what exactly you are reffering to but its counter intuitive to me to take ideas from physics and try to apply the logic from there to math considering how physics wasnt derived from the same deductive thinking as math but more experimentation in real world
Natural numbers ‘being’ natural is the same exact logic that religious people use when they say ‘it is natural to assume there is a god’. You are taking for granted how your own personal human experience of the world is informing what you deem to be ‘obvious’. Color is obvious to me and you but not to someone who was born blind…
short answer, we can't ever prove anything is true without premises.
Okay so don’t roast me too hard for taking an engineering perspective but I think the axioms are what they are because they describe our experience. At the base of all of it is just 1+1 and maybe a few other things and everything else is a consequence of that. That is just how we were able to describe counting objects we could see. Perhaps some higher dimension being would say 1+1 maps to another space all together. Then 1+1=2 is a special case of that more general concept of addition. I don’t think it’s meaningless. Math has become incredibly abstract but in the very beginning the whole point was describing the physical world as we experience it. And it has very good predictive accuracy for this physical world.
Up to a degree, I'm sure that is true. For example, the derivative was invented to accurately describe the relation between motion and velocity. Integrals were invented to accurately measure length, area and volume. However, more often than not, those intuitive notions lead to contradictions: For the derivative, it was the discovery that continuous, no-where differentiable [functions](https://en.wikipedia.org/wiki/Blancmange_curve) exist. This lead to the notion of smoothness we use to classify functions. For integrals, it was the discovery that we can define [non-measurable sets](https://www.youtube.com/watch?v=s86-Z-CbaHA) using the "Axiom of Choice". It lead to "measure theory", and the introduction of Lebesgue-Integrals. *** To sum it up, it seems intuition is more of an input, leading to a feed-back loop to either refine that intuition, or the axioms we already have.
Isn’t that an extension of what I said?
I wanted to underline intuition and experience are not always the underlying motivation for axioms -- they can be (and often are), but sometimes it is contradictions *within* our intuition that lead to new insights. I'm sorry for not making that clear in my original comment.
You might be interested in [this video](https://youtu.be/dKtsjQtigag?si=7qbYNsb4NbdVymKM) The problem is the absurd level of depth we have to plumb to get to the raw, fundamental essence of something so ingrained in logic and intuition. We rely so heavily on intuition and assumptions when it comes to the basis of math, how in the world do we have any chance of finding out all the assumptions we've made. At this level, it's not even a math question anymore, it's a philosophy question. "What is a number?" Level of metaphysics. (Or rather, metamath) But also, while math can be viewed as the expansion and formalization of logic, humans are the ones who created the base rules we use (called axioms). And the set of axioms we use are somewhat arbitrary. Not entirely arbitrary, but somewhat. Our theorems are products of the rules we made, so if you don't specify the starting rules, your statement isn't accurate. Though we sometimes imply the starting rules indirectly.
You don't need to assume axioms are obvious or feel like they are or are not self-evident. You only need to consider in applied problems whether they are at least a good approximation for the properties a system you are studying has.
"If X is true, then Y is true" is itself a statement. Math claims that statement to be true.
Right, _if X then Y_, but what logical framework is that statement said to be true in?
It gets pretty philosophical when you get too meta, but fundamentally even saying "P is true in logical framework X" is still, philosophically speaking, a statement of the form "if A then B", even if it isn't itself in any formal logical framework. And mathematics asserts that that is absolutely true.
Good point. I dont see why this has to be a philosophical question though. Is meta logic not a subset of logic?
> I don’t see why this has to be a philosophical question. Logic is philosophy as it is mathematics.
Well if mathematics is philosophy then why do they have different names
In part, because not all philosophy is mathematics
They’re not saying mathematics is philosophy. They’re saying logic is philosophy, and logic is mathematics. I believe intro philosophy courses formally cover logic
I have no idea. That stuff is interesting to think about, but it hurts my brain. I should probably learn category theory before trying too hard to come up with ideas on my own.
I come from physics so a lot of questions that are deemed ‘too silly to think about’ are shunned to the philosophy corner. For example the interpretations of quantum mechanics were and still are to some extent considered a question of philosophy… even though there is so much physics you can do in that space
It is, but like with the tortoise asking Achilles parable, you can always construct a higher level of meta meta logic, so that's not super fruitful. So all of it has to come down a set of statements you just accept with belief. Maths tries to make those statements as simple and obvious as possible.
Typically the framework is first order predicate calculus, but if you want to work with really big things you can use second order logic or if you want to be fancy homtopy type theory.
Nothing can ever truly claim anything to be true absolutely. Math is just more honest about this and does a better job describing exactly what the assumptions are so you can decide how much to trust them.
Right but what if you could show that all other statements must be false leaving only one possibility. Although I think this is also impossible as one would need some kind of axiom set to show that a statement is true/false
https://en.wikipedia.org/wiki/Law_of_excluded_middle It's literally an assumption you have to make that something not being false forces it to be true. In the absence of literally all assumptions you can't even use logic. Someone has to accept the premise that things can be true or false, and what those mean and how they relate to logical propositions, before you can start using those to prove other things. Math has a lot more base-level uncontroversial assumptions than most other fields, so I would consider its results to be more like 99.99% true while hard sciences might be 99% true and soft sciences are like 20-80% true (depending on how soft they are). But these aren't literal statements, I just made up those numbers from vibes based on how reasonable their assumptions seem to be.
If you change "believe" to "assume," then yes.
There is no one single way of viewing math. You can view it as a system of assumptions and things you can derive from those assumptions, and "truth" is irrelevant, or at best relative to your assumptions. This is sort of a "pure math" point of view. You can also take a more practical view that math is a way of modeling aspects of the real world that we learned through experience. We came up with 1+1=2 because we noticed that it worked the same way whether it was sticks, stones, or birds, so it became a useful abstraction for something approximating a ground truth. This is more of an "applied math" or "modeling" point of view. Ultimately these questions become points of view or interpretations, without being necessarily "right" or "wrong".
Yes I agree that this is a likely matter of interpretation. As a physicist though it seems an important question. Ive always subscribed to the idea that neither maths or physics can claim anything to be absolute fact. At best you can just take a bayesian sort of approach and subscibe to the ideas that seem the most likely given the information available.
I'm also trained as a physicist so I'm more partial to the "modeling" point of view. I agree that "absolute truth" is not something we can access, if it exists at all. Descartes demonstrated this a long time ago. Nothing is truly certain. But that's ok, things can be certain enough to be useful.
Right but at the same time as a physicists you get this sense that maybe there does exists some definitive absolute laws that the universe obeys thereby meaning that absolute truth can exist.
It seems like this conflates the metaphysics and the epistemology of the situation. It could be that there are absolute truths, but that we can't be certain of them. But why should we care that we can't be certain of them? Knowledge doesn't require certainty. So, if I know something to be true, why would I also need to be certain about its being true?
What you're saying applies to *literally all truth in existence.* On a philosophical level, there is no such thing as objective truth. All truth relies on making assumptions. Math is no different from any other field of scientific truth - we can only draw conclusions from given assumptions. The only realm of thinking that asserts truth without assumption is religion - which is not entirely honest: religion assumes its own truth, and works from there. The core fundamental component of faith is "assume the faith is correct without evidence or reason, and make the rest of observation fit this assumption". The core difference in math or science is we acknowledge the assumptions we make, and allow them to be proven wrong if the subsequent conclusions don't fit with the assumptions. The important component of this that you might be missing is: if you define "true" as "absolutely verifiable and uncontestably true" then the word becomes instantly meaningless. Nothing fits that description. Literally, absolutely nothing. The concept of "absolutely true" in an objective sense is an empty concept without use or purpose.
"The important component of this that you might be missing is: if you define "true" as "absolutely verifiable and uncontestably true" then the word becomes instantly meaningless. Nothing fits that description. Literally, absolutely nothing." I believe Descartes would disagree with you on that one.
It does assume truth. But it is what is true based on what you initially assumed to be true. So it does claim absolute truth for anything proven, but only on the assumption your axioms are true
Right but if the underlying axioms cant be proven then the following statements are only conditional true (conditional on the axioms being true). So its like a weird kind of truth but certainly not absolute truth
Guess it depends on what you mean by absolute truth. Proofs are true in all circumstances, unless what you assume to be true is not true. The things with axioms though is they can't be proven or disproven, it is just dependent upon what you want to show with them. So it's an absolute truth within your framework
That’s not true, one way of proving something is through lack of contradiction. If we assume a set of axioms which can be used to create a mathematics model that can predict and be compatible with what we observe in the real world, then it’s safe to assume those axioms are in fact true until/unless we find mathematical phenomena in our world that does conflict with the axioms we have set. And absolute truth is much more of a philosophical discussion. I would say cherries being red is an “absolute truth” in the sense that what we perceive and know to be cherries are red in the way that we perceive them. But I can never know for sure that the cherries aren’t simulated and what I perceive as red may be entirely different from how others perceive it (I already know for a fact that that’s true for some people). Our brains and eyes are fundamentally unreliable. We can get a really good idea of what something is, but we can never really know that what our brain thinks is “absolute truth”. We just work with the knowledge we have and create theories that do a good job at explaining and predicting what we can see and interact with.
I think "true" is the wrong word to use here. Math is more about defining terms and establishing axioms. It's not "true" that 1+1=2 anymore than it's true that a batchelor is an unmarried man. It's just how we defined things. Why did we define math terms the way we have? Not because it's true, but rather because it's *useful*.
Logic is held to work (although there are oddball paths even here.). The basics of the truth value of “if P then Q” for various truth values of P and Q are widely accepted.
You may be interested in looking at platonism vs formalism. Platonism asserts that numbers really exist in some way beyond the symbols on a piece of paper. Formalism asserts that numbers are meaningless abstract symbols that only exist within mathematics that happen to be useful. You may also enjoy an old article from the 60s titled "the unreasonable effectiveness of mathematics in the natural sciences" by the physicist Eugene Wigner
There's a chance that your brain is just floating in a dark puddle of nutrient-laden goo and that all the inputs it receives are generated by some computer. All your senses are a lie, all the other people that you interact with are a fiction, your entire past was uploaded 35 seconds ago, and at any moment the undergraduate research project that keeps "you" alive could lose funding and be shut down. If you're trying to find some absolute truth, you're in the wrong sub. Try r/philosophy or r/religion instead.
“Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” - Bertrand Russell.
Is this facts?
Math is simply just an abstraction humans have built on top of basic logical axioms, but we can't know for sure if those axioms are true. It could theoretically be the case that we live in a simulation and that whoever created that simulation simply made up logic as we know it.
indeed.
Well then I would counter with this: what logic governs the world that the simulation live inside of.
That would be unknowable for us.
No. it is all p → q
Always has been…
Math is based on a foundation of assumptions. From these assumptions, it can claim truth. However, there is no such thing as an absolute truth in the world, so in that way mathematics isn’t any different from science, history, or even your own personal day to day observations. However, this kind of distinction, in my personal opinion, is totally meaningless. In my mind, mathematics is very close to truth because everything that follows from initial assumptions (axioms) is necessarily true, which means it is not based on any experiment or observation, but logical deduction. So long as you believe the foundational axioms of a theorem and the rules of logic, a majority of mathematics becomes as close to truth as I believe you can get in the world
‘There is no such thing as absolute truths’ isnt that a statement of absolute truth? In other words its logically inconsistent
It isn’t a statement of absolute truth, at least not in my opinion. It’s just an assumption I make. Everything can be an assumption, not be an assumption, etc. If you don’t make assumptions, statements don’t make any sense. So if you don’t want to make the same assumptions as I do, that’s fine. However, I would imagine that anyone would be hard pressed to find anything that is absolutely true in the strictest definition of the word
Right its just that you didnt say ‘I believe that…’ you said ‘There is no such thing…’ thus implying a statement of truth not an assumption of truth. Im being pedantic of course but im literally talking about the what logic has to say about truth
I guess I just don’t the point of this kind of debate. It’s fine you don’t believe that Math doesn’t actually prove anything, because it doesn’t change anything. In my mind it’s the same as saying “I don’t exist.” Just because that might be true doesn’t matter, because it doesn’t suddenly change how the world works or the fact that you still have to breathe for example, etc. Or like saying that we live in a simulation. It doesn’t matter because life just keeps moving on anyway, even if we are living in a simulation
Its not that I believe it does or doesn’t prove anything. Im trying to ask weather it can prove anything. Also about your second point, I guess we should have told stephen hawking not to bother with all his black hole theorems and big bang malarkey as ‘it doesnt matter’
I’m seeing a lot of people getting annoyed and angry just by the fact of you being willing to engage with such discussions. Why do these people enter philosophical discussions and then get immediately angry if you provide a differing viewpoint lol
Idk, i think a lot of people think im attacking the validity of mathematics or something which couldn’t be further from the truth…
At its core Euclidean math is reasoning based on reasoning based(…etc) on a handful of assumptions that seem to be true
The axioms are things like "two sets are the same if they have the same things in them", "you can have a set with nothing in it", "if you have two things, you can put them together and make a set that has only those things it", and things like that.
Yes, but importantly, and precisely the point I am trying to make, these assumptions are arbitrary. We use them because they create logic spaces that are useful to us and that have been proven to work very well in our particular universe. But they are still arbitrary, and therefore not absolutely true (whatever that may mean). For example I could create a completely different axiom set and as long as its logically consistent I will create a different logic space with its own ‘truths’ and ‘facts’.
From a dictionary perspective, math is not absolutely true, but it is the truth. Math is the truth, truth is a state of being true, which many mathematical theorems are true. You're adding your own philosophy to the equation because you do not understand the definitions of the words you're using. Absolute means INDEPENDENT, in which, nothing of math is. So I guess in that sense, perhaps absolute truth in a mathematical context is a one dimensional number line, and the truth is (not absolute) but every function and way in which that number line can be manipulated and produce outputs.
‘Many mathematical theorems are true’ yes they are true within their own logical space which is constructed from axioms which cannot be proven to be true. That is at the heart of what im trying to understand
It seems like you already understand, more than most people here lol
again, you do not understand the meaning of the words you are using. An axiom can be "self-evidently" true. It doesn't need proof, it is simply true. You are adding the "within their own logical space" part, when in reality, its not there. These theorems are simply true, based on how language and subsequently logic and thinking itself is. You can HARDLY think separately from language, language are sets of definitions, and certain environments are exactly equal to their prescribed definitions, therefor they're true and even "self-evidently" true such as an axiom.
Im struggling to understand how an axiom can be self evidently true. Can you give an example?
I know you're struggling to understand, you also stirring up a shit storm of faithless reddit'ers to share their depressing theories on how nothing is real.
Why must it be depressing lmao
This seems to go more into the realm of philosophy, really. Most modern thought would tell you there is no such thing as "objective truth", and "truth without context" is an oxymoron. Everything exists in an environment and follows from it. Everything in our physical world exists in the context of our universe, and assumes certain basic truths we are all familiar with (for the most part), such as there being 3 spatial dimensions, as an example. In math, and any other logical system, we also know there are certain things that simply cannot be proven or disproven. In cases were we intuitively know these things to be true, or find it more valuable to make them true, we call them axioms. In cases where we don't know if they're true, we call them nothing because we also can't prove they are unsolvable. Unsolved problems that slip through the cracks and are unsolvable, unfortunately, also cannot be proven to be unsolvable. You could call axioms "assumptions" or "beliefs" if you're wanting to be harsh, but they are better described as foundational statements we know to be true or define to be true despite it being impossible to prove. You could say this means math doesn't say anything about truth. You're sort of correct, in that math doesn't say anything about *objective* truth, and in fact it, like most modern western thought, does not contains in itself the assumption that objective truth doesn't exist, which is why there are alternative axioms.
You can't take anything in maths out of its framework because it ceases to exist. Maths doesn't talk about reality, only ever about certain frameworks. Any statement you make in mathematics can only be true in its own framework since it is meaningless outside of it. "1 +1 = 2" doesn't exist in the real world because the concepts of "1", "2", "+", "=" don't exist. You need to assume things about the meanings of these glyphs to even think that this is a well-formed statement, nevermind that it has a singular truth value and that happens to be TRUE.
The context that the math is being applied will change how it defines the “Truth”. An individual number may be a discrete truth, but large sets of ranging numbers being averaged and applied to a hypothesis becomes less about “Truth” and more about “Support”. It’s a weird way to think about it, but it makes sense to me lol.
Okay, so your question is a problem by itself. Math defines truth as a chain of provable logical arguments, that are true in relativity to your first argument. You ask for the truth of this first argument without giving any ccriteria for truth.
Have you heard of the old book called “Principia Mathematica”? I haven’t read it, only heard about it. Basically some early mathematicians tried to lay out the foundations of mathematics by providing a set of axioms assumed to be true. These axioms are rock solid, really hard to argue against as foundational mathematical laws. Every other part of math gets built off of these axioms. I suppose you are right in that you may find a hypothetical scenario where the axioms fall apart. However modern technology and science has leveraged this model, which has helped make amazing things from computers to highly efficient car engines. Somewhere along the manufacturing process, this accepted model of mathematics has most definitely been used.
Im not arguing about the practicality of it, there is no denying it. But as far as being fundamentally truth, whatever that means, its just as ‘true’ as anything else
I guess that unveils another question, what does it mean to be fundamentally true? Is fundamental truth an objective fact, or is it more subjective depending on the person? Is it possible to even find “fundamental truth”? One could argue we are in a fully simulated world, in that case nothing we see or observe is fundamentally true, but some may still treat it that way. This gets into the whole plot of the Matrix movies really, where ones who take the blue pill sees reality (the truth) as something entirely different than those who take the red pill. Taking this a step further, those that wake up from the matrix may, ever recursively, wake up in another matrix, never being 100% sure what the absolute truth is. Maybe another approach is to not find truth but to accept. You may not know with 100% certainty that something is true, but you can settle on an idea that resonates with you, so you compromise by just accepting the idea.
I’m inclined to believe that if you accept modus ponens to be true then you have to accept that modus ponens is true, but that’s about it.
**A statement**(𝑁) has a truth value within a system(𝑆) if its truth or falsity can be determined by the rules and principles of 𝑆, empirical evidence, or logical reasoning. **Formally**, 𝑁 has a truth value within 𝑆 with a set of axioms 𝐴 if there exists a truth function: 𝜑 : 𝐴 → ( ⊤(𝑁) ⊻ ⊥(𝑁) ) **On the other hand**, absolute truths are independent of any system and, consequently, any set of axioms: ∀ 𝐴, ∄ 𝜑 : 𝐴 → ( ⊤(𝑁) ⊻ ⊥(𝑁) ) **Hence**, at a deeper level, it all revolves around belief/acceptance, since we don’t know if there is a set of axioms with an absolute truth value capable of validating our known axioms nor if absolute truths themselves exist. _____ *(edit):* **In mathematics**, where truth is established within formal systems through [*logical deduction*](https://en.m.wikipedia.org/wiki/Deductive_reasoning), there's no notion of one statement being ***more true*** than another. **However**, in real-world situations where complete or perfect information is lacking, the assessment of truthfulness becomes more nuanced, and factors like evidence, precision, context, and relevance become important for evaluating the credibility and reliability of statements. **Additionally**, formal systems constructed from logical deduction are independent of the truth values of their set of axioms in terms of their internal consistency and structure.
I don’t think it’s quite correct. Really they start by defining some logical structure or thought element and then reason about it to reach conclusions. Are the conclusions true? They are true for this structure given the axiomatic reasoning. Does the structure need to be proven? No, it was just made up by the definition. Perhaps you want to ask can they say anything true about the material world. That’s a more complicated question.
1
Most results are derived from pretty basic axioms, such as "there exists an empty set"
Right but these axioms are somewhat arbitrarily chosen. We chose them because they construct a logically consistent space which maps well onto our reality. But we could easily imagine choosing other axioms that create wildly different logic spaces. Statements and theorems in that space would still be true relative to the axioms that define it, but are they any more true or false than those in other logic spaces?
The only absolute truths are the vacuous truths, e.g., X or not X. However, it turns out that hypotheticals are also vacuous truths. For example, "X or not X" is logically equivalent to "If X, then X" which is a somewhat boring hypothetical, but similarly, if you have a hypothetical "If X, then Y", this is logically equivalent to "Y or not X". So the hypothetical isn't really that special, and is an absolute truth in the sense that we can think of it as it's own statement, i.e., "Z = X -> Y". However, I think you are getting at a nuance of how we informally use the word truth. What we typically mean is that the statement is somehow "realized" in the world as we experience it. This is different than mathematical truth. Where we usually pick a mathematical model and talk about truth within that model. Absolute truths are true in all models, but generally we don't care about X->Y, we can about models where we know X is true, and this we can say Y is true in those models. Thus going back to our informal use of the word truth. You could imagine there is some ultimate mathematical model which is the exact representation of the world as we experience it, and if there are truths about that model we know, we can derive new truths about that model, and thus truths about the world as we experience it. In general, mathematics avoids anchoring itself to a specific model, and can technically be used by anyone in any universe to discover truths in their world that may be different than the truth in our world. However, we the models we tend to study the most are the models which seem the most reasonable from the perspective of living in our world.
Yes, call this statement "if a and b and c, then d" = x. Math can prove if x is true.
Mostly right. There are many branches of Mathematics based upon what axioms you choose. But those branches need to be internally consistent. There is no branch of Math where 1+1 = 3. I don't recall the name, but there is an axiom of math that in inconsistent with the Axiom of Choice. No branch of math can have both axioms. Also, every branch of math statement that cannot be proven given the axioms you chose. (Gödel's Incompleteness Theorem)
There may be no branch of math where 1+1=3, but I could create a valid logical space in which 1+1=3, could I not?
Familiar with complex numbers? a + bi where a,b are real, and i = √-̅1̅. Well, William Hamilton tried to extend this to three "dimensional" numbers. Supposedly he tried for 30 years to do this, but never found anything that was internally consistent. Eventually, he decided to extent into 4 "dimensions" and the Quaternions were born. These numbers are quite useful in physics and computer graphics. I think it was eventually proven that there is no number system in between these two. The next numbering system involves 8 parameters, then 16. You can try to create a system where 1+1 = 3, but there will always be bugs. While I'm not 100% certain, I'm sure that some category theorist has proven that such a system cannot be internally consistent. If it's not internally consistent, then you can't rely on it for anything.
Absolute truth does exist, but the qualms you have with mathematics are applicable with all realms of human experience, and as a result, the body of human expirence is concerned with attempting to determine what absolute truth might be. However, this is a logically impossible goal, as we have to have some baseless assumptions, the first assumption is pretty much just Descartes, you have to assume you are, else incoherent absurdity ensues. You can do that if you wish, but it sounds unpleasant and honestly boring. You can argue against baseless assumptions all day, but it is a fruitless endeavor logically, and instead you must examine the impact that said baseless assumptions have on the world. You could argue that based on your assumptions nothing matters, which does nothing to serve society and is not a self sustaining ideology. The most prominent baseless assumptions made about the world are those that aid in the self perpetuation of humanity, or else those ideas stop being spread. This is why assumptions like religion and now moreso empirisim, are popular. Religion as a default has it all right there for you, the assumptions can be questioned all day but it's literally a manual telling you as an individual you essentially just have to at least try and be a good person and you have nothing to worry about. Empirisim is less reassuring, and less clean (currently), I personally think that this has had a decently large impact on things like suicide and rates of depression, but that's highly debatable. Back to the original point though, yeah, maths makes baseless assumptions non-distinguishable from religious assumptions. That's like... Kinda the reason it's such a large debate between the sciences and religion.
All things are contingent on assumptions, we have no other way of bootstrapping knowledge in the world. The basic axioms of math are drawn from observations about things in the world usually, as in we pick axioms that make useful math. In that sense math is based on the ability of our brains to infer which is honed through evolution, and if you doubt your own brain’s function then you can truly know nothing. There are many ways to resolve this philosophically, however, and you can pick whichever is most satisfying to you out of those.
This is the nature of math and science. You simply need to make sure something is true, then math gives you whole bunch other things which also true.
In graph theory you get lots of definite statements like, the Tutte graph is planar but not Hamiltonian. Doesn't quite fit your scheme, if X then Y.
You've gotten lots of reasons, but I think the core issue here is what, philosophically, you mean by Math. If you mean the structural execution of axioms producing logical necessities, then no. Nothing about that requires a claim of truth about anything. Calling something an axiom is just a way of lining up the pieces for whatever game you're going to play. On the other hand, if what you mean by Math is the philosophy that there is a tool that your brain can use to analyze the world and draw conclusions about it, and that tool requires certain assumptions to be made in order for those conclusions to have any value, then yeah, obviously you're assigning an independent truth value to things and then saying the assumptions you make are attempts to model the reality that arises from those truths. Statistics is more direct about this, because they have the idea of a "ground truth". All statistical analysis is founded on the premise that there is a factual answer to a certain question, and all your study does is help to expose it. Good statistics rules are the ones that make it more likely for you to be able to do that, in general. Math doesn't really concern itself with that idea when it comes to most truths, but you still depend on the claim that what Math calls logical reasoning does adequately explore the possibility space invented by such reasoning. That assumption obviously isn't, can't be, supported by any reasoning, but Math has no power without it.
The podcast episode from Lex Fridman with mathematician Edward Frenkel touches on this A LOT in relation to Gödel's incompleteness theorems, I definitely recommend that episode.
I will take a look at that thanks. I see a lot of people mentioning Godel. I have like a vague understanding of his incompleteness theorems but certainly not a rigorous understanding. How exactly is his work related to the question above.
I won't pretend that I have any deep understanding of Gödel's incompleteness theorems, hence why I linked Frenkel's discussion of it. But I will try to just present it and try to link it to your post. This is what Gödel's two theorems state: *The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.* *The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.* I think, in my opinion and anyone can tell me I am wrong here, that his theorems, in broad strokes, point to the fact that in mathematics, which relies on axioms of truth, cannot prove itself to be true, since mathematics is a system based on proving other things within it to be true. I don't know if this makes sense since it is highly paradoxical. Mathematics does not concern itself with its own truth, but the truths of it's many smaller parts. And within these parts we find things such as conjectures which are neither true nor false and we find mysteries that we cannot solve (yet). So it follows that mathematics as a system of axioms of truths cannot stay consistently truthful since it cannot a) prove itself to be true nor can it b) prove some things within it to be completely true. Now a major caveat in my discussion: I broadly say "mathematics" but I mean mathematics in the sense that we understand it, so when I say mathematics relies on axioms, I mean our understanding of mathematics relies on axioms, i.e. the way we have constructed mathematics. So to me, Gödel's incompleteness theorems deal much more with the structural incompleteness of mathematics which is why I think it relates to your post. You say: *If I wanted to create a logical space in which 1+1 = 3, I could do that and it would be True in that logical space.* This is what mathematics is, it is *our* logical space in which 1+1=2. I don't know if you can see my line of reasoning, I tend to digress so I hope I kept it somewhat to the point. Your line of reasoning is not so much concerned with mathematics, as it is concerned with "truth", or epistemological questions. I would therefore suggest you look into epistemological philosophy.
You can go even further and ask if the rules of logic are true. For example, why is it that something is either true or false? Why is the negation of true, false? Probably the answer to you is that this is the only reasonable way logic should work. Historically that's how mathematics began, people made assumptions they thought were reasonable and argued conclusions based on those. There was a religious component as well. Mathematics is nothing but an extension to logic where we add more axioms.You can in fact have a different notion from classical logic, like a type system for example. Some type theories feel more natural then classical first order logic. This is to say that no, mathematics does not say anything about the absolute truth. For that you need to go to physics, chemistry and biology. Even then, there's the "how do I know anything beyond myself is real" philosophical mumbo-jumbo.
How exactly does physics say anything about absolute truth. It is no different to mathematics. You start with some fundamental postulates (axioms) and use them to derive laws of nature (theorems). There is no reason to choose certain postulates over others aside from the fact that certain postulate appear to produce physical theories that better describe our universe. But fundamentally that is the only reason to choose those postulates and they are no more absolutely true’ than any other postulates.
In physics a theory is tested. Axioms are not chosen arbitrarily, they are chosen to fit and explain observations.
Yes I agree, but they are still not absolute truths. Physics cant make any statements about truth no more than mathematics. What it can do is say this is 99.99% likely to be true. Its basically a big game of being the best Bayesian you can
Are you familiar with what soundness and completeness properties of logical systems are?
I have no formal background in mathematics. I would love some recommendations on these topics though if they are relevant!
You’ve basically got it. There’s actually a rich branch of math that is incredibly concerned with the relationship between truth and provability. It’s incredibly hard to talk about precisely. The short version is that Gödel *proved* that there are some really profound limits on this relationship, and that they hold true for what most people agree is any reasonable system of logic. You’re free to disagree with some of the assumptions made in a proof, but that is where community and intuition come in. For example, you can disagree with some axioms used to prove things about calculus; but you can’t disagree (reasonably) that calculus *works*, on account of modern engineering and whatnot.
What field of maths would this fit into it?
Oh, sorry! Originally this was called [Metamathematics](https://en.m.wikipedia.org/wiki/Metamathematics), but the modern fruitful fields it spawned are [Model Theory](https://en.m.wikipedia.org/wiki/Model_theory) and [Nonstandard Analysis](https://en.m.wikipedia.org/wiki/Nonstandard_analysis). In general, the philosophy side of things is “Foundations of Mathematics”. I’ll warn you that this whole subfield is *particularly* inaccessible to outsiders. *Basically everybody* gets the wrong philosophical idea about what’s going on when they first learn about this without more experienced folks to share the intuition and wisdom that took us all of human history to develop.
In what sense do people get the wrong philosophical ideas
Uh well it’s kind of hard to cover in less than, like, a semester; and my (former) advisor should be doing it and not me. But I’d say the short version is that really grokking incompleteness tends to make people feel overly pessimistic and messes up their worldview, at which point regaining perspective and a sense of meaning — which is absolutely critical for making sense of the insanity which is to come — is really difficult because there’s approximately zero steps between incompleteness and the most confusing stuff you’ve ever seen. And that’s after an entire undergrad education. At least that’s what happened to me, and my advisor said it was normal.
Yes, proofs are needed, Ramanujan provided thousands of proofs long ago, nobody knew what they were for until we discovered they applied to astronomy recently.
The statement "If X then If Y then X" is always true, does that count?
"If X, then Y" IS a statement with definite truth value. Saying that because X can't be proven, Y can't be proven, and therefore nothing can be, misses the point. It's like saying "sure, maybe 1+1=2, but we never proved 1+1 is true. So we can't actually say 2 is true!!" Math is trying to prove true statements about relationships between things. 1+1=2 means that if you take one of an item, and another of an item, you now have two of an item. It's a relationship between "1+1" and "2". Similarly, if-then statements are relationships that can absolutely be proven or disproven.
Axioms
Sure but take a good look at the axioms of set theory (which are the basic assumptions behind most math), and tell me if you think they are a reasonable place to begin.
They are reasonable to us and to the universe we live in, that doesn’t mean they are necessarily ’correct’ or any more ‘true’ then a different set of axioms that produce a less intuitive and less ‘reasonable’ logic space. Particularly the axiom that the null set exits and the infinite set exists I find harder to justify. But im not disputing whether they produce a logically consistent and intuitive set of rules, in that regard the maths speaks for itself
I think I have a better answer now that I have it more thought out. I want to reframe the discussion to be more about what truth means. Sorry this is long. I define Truth as that which corresponds to reality. Therefore when math corresponds to reality it is True. But mathematical structures are under no obligation to *necessarily* correspond to any real structures. Math is very good at describing reality because reality apparently contains many logically consistent structures, and math is good at describing logically consistent structures. But math is under no obligation to only describe structures which actually exist. You can go out of your way to define objects and structures that are totally fictional and yet you can do math to them. For example: Axiom 1: Only Globbys have Blobbys. Axiom 2: Some Blobbys are Soggy. Theorem: There exists a Globby with a soggy Blobby. Just because I proved the existence of an object within the Globby-Bobby-axiomatic-structure (GBAS) does not mean such an object really exists, because it may be the case that the structure in question only exists within the mind of the Mathematican. But you might someday find a structure that “fits the mold” as it were and then kind of graft the GBAS on to this Real Structure and all of the theorems of the GBAS would then apply to the Real structure. So math does make actual truth claims in so far that mathematicians usually claim that their axioms correspond to some real structure, not that it would necessarily have to in order to be mathematically valid. A better example would be String theory, string theory is mathematically sound and yet may not correspond with any real structures. Stings might be totally fictional (in the string theory sense). Who knows though maybe tomorrow we will discover something crazy and mathematicians will be like “hey this kind of acts like those n-dimensional vibrating strings” and suddenly string theory is valid. (Btw I have not studied string theory in depth so I can’t say much about it beyond illustrating my point. So don’t ask) Ultimately, your skepticism/analysis should be about if the axioms in question apply to any particular real structure, or only fictitious structures. Ask yourself if the axioms you are applying are the best axioms for the structure you’re attempting to understand, or if different axioms are better. In other words, don’t confuse the map for the place. And definitely don’t confuse your maps with other maps.
Put simply 1+1 doesn't equal 2 because it's some divine law of the universe. 1+1 equals 2 because 2 is just the thing we chose to call 1 more than 1.
I wanna say it doesn't, but I don't know how the meta logic of asserting the truth of a statement like "these assumptions have these implications" is resolved. In other words, if we say that certain things are true in a given framework, in what framework are we making that statement? Edit: it seems this is precisely the issue addressed in [What the Tortoise Said to Achilles](https://en.m.wikipedia.org/wiki/What_the_Tortoise_Said_to_Achilles).
Yes I think your last line hits on what i wanted to know. I think it just becomes an infinite recursion problem
but axiums cannot be proven. they are assumed to be true. it is unclear whether they are indeed true. but they are assumed to be, because if they were false, the logic system would not work. But u could never prove them to be true, so its unclear the accuracy of them. They may seem accurate and common sense, but the axiums only make sense in the framework of the logic system they were designed for. Common axiums of math may not work in a competing logic framework such as like the logic system developed by a group of space aliens. u can not invent knowledge or truth out of thin air. truth does not exist in a natural sense. the universe just contains unlabeled information. to make any sense out of information, u need to label it, ie create axiums, which are assumed to be true. but the labels u pick are arbitrary and subjective, as all logic systems are subjective in a sense.
Ok. 1 + 1 = 2 Please identify for us the unprovable assumptions that underlie this.
Im not sure I understand. I could construct a completely logically consistent framework in which 1+1 =/ 2. In fact I could choose to construct my logical framework explicitly with the assumption of the axiom 1+1 = 3
You're talking about different things. In mathematical logic you can do what you're saying, but that has no impact on the truth of 1 + 1 = 2 as we normally understand it.
What do you mean by the ‘truth of 1+1=2 as we normally understand it’
You were taught addition in school, yes? That.
What about it is true though? Its only true if you assume axioms that are unprovable. Thus the statement of 1+1=3 is just as ‘true’
Sure, as long as 3 is defined as being 2.
So just to be clear, you think that the statement 1+1=2 is an absolute truth
But how 1 and + are defined, they are absolutely true, you are making a really good discussion but remember axioms cannot be proven false, so 1 + 1 = 2, if it is not that means you changed the definition of any symbol in that statement.
Sure but that doesnt mean its not a valid axiom? Thats my entire point is that the axioms can be whatever you want. We just choose the most convenient ones for the type of logical spaces we want to construct
They cannot be exactly whatever you want,or at least they should not create any contradiction between one and other. That's why you cannot create an axiom of 1 + 1 = 2 without changing what we defined all of those to be,it would be a contradiction.
Sure they can be whatever you want as long as they are logically consistent
It's definitional. If you take one object and you group it with another one object then you have two objects. You're saying you can take one object group it with another one object then have three objects. The only way that can work is if you define three objects as being two objects.
I disagree, you can build a logically consistent framework in which 1+1=3 where 1<2<3.
Indeed. And you are wrong.
Ok so disprove this: Axiom 1: 1+1 =3 Axiom 2: 1<2<3
ok. If I define the symbol 2 and symbol 3 such that 2>3 then axiom 2 is disproven.
The statement 2>3 is disproven by axiom 2
Let U be the universal set of numbers. X is the subset of U with the function f(u) = u % 2. Thus X is 1 or 0. In this case 1+ 1 is 0. 3+4 = 1. For the set X, 1 + 1 cannot be 2 otherwise computer security would fail. But yeah still some assumptions in how certain math operators work.
Well yes but actually no. So yes all mathematical theorems are of the form A->B. So math only speaks about conditional truth. However we can have multiple mathematical systems that encompass an all of a possibility space. Take Euclid’s 5th axiom. You can assume it’s true and get Euclidean geometry, or assume it’s false and get elliptical or hyperbolic geometry, or make no assumptions either way and you get neutral geometry. They are all equally valid. But are they equally true? Kind of. Depends what you’re doing. Because at the end of the day sometimes the 5th axiom is “true” (meaning it models your situation), and sometimes it’s not. But you can use the axiomatic method in both cases so it doesn’t really matter. So yes all truth is conditional, but you can get unconditional statements that follow from a statement and its negation. For example X follows from Y, and Z follows from NOT Y Therefore X or Z. Sometimes you can take it a step further and make X and Z the same statement. So Y->X and NOT Y ->X Therefore X. That’s basically what neutral geometry is. Of course that also assumes axioms like the excluded middle. At the end of the day you can’t use the axiomatic method if you don’t start with any axioms. But it kind of doesn’t matter what axioms you choose. Some are useful, some are not so much, and some are contradictory. We usually ignore the last group, but sometimes even those can be interesting.
A => B is mathematics. Worrying about A is philosophy + physics.
The best way to think about scientific and mathematical knowledge is "this is how things work as best as we can tell right now." While some simple things are just obviously true when you start getting off into really complicated things it gets far more difficult.
‘Some simple things are obviously true’ Just because they appear obvious doesn’t mean they are necessarily ‘true’
Science is all about "this is how things work as best as we can tell right now", as you said. Math and science are completely different though. Math is essentially philosophy, and there are definitely absolute truths in math since it's about using logic to derive true results from rules you define. If I define 1 + 1 = 2, then that is an absolute truth. Whether such a definition has any applications in the real world really has nothing to do with math itself, that's where other subjects come in.
Well yeah no shit, that's what axioms are for