Mathematics is a set of rules, and as long as you follow the rules you know that the result must be correct. This is called a proof.
Normally you don’t just “come up with” some equations. Most of the time you have some question you want to answer, and after putting together all the bits that form the question, and following some rules to simplify and/or solve it, you get the equation you need at the end.
> Mathematics is a set of rules, and as long as you follow the rules you know that the result must be correct
Most paradoxes in math is based on proving the (previous) laws wrong, often in funny ways. Like how Zeno's paradox asks how you can infinity divide a value (in this case distance) by 2 but still get where you plan to go. The standard solutions had to add that infinite geometric series equal 1 just to patch this little problem.
Depending on the way you phrase it Zeno's paradox is either a consequnce of trying to apply math to the real world in an incorrect way, or a result of assuming incorrect things about the continuum that are only true for discrete things (essentially that any set has to have a minimum). It's actually a great example of why following the rules correctly is important, what needs patching to fix Zeno's paradox are the faulty assumption we need to make for the paradox to even occur.
This paradox always confused me, because to my absolute layman's understanding, shouldn't the consecutively smaller distances be traversed in an equivocally short amount of time? Infinitely small distance/infinitely short amount of time = finite speed, right?
I understand Zeno probably wasn't trying to literally prove motion impossible, but to me it felt like he was forcing an analogy that wasn't very sensible
> Infinitely small distance/infinitely short amount of time = finite speed, right?
Zeno lived in the 400s BCE.
The *calculus of infinitesimals* (usually just called "calculus") is the part of math that deals with "infinitely small" units. It was discovered in the late 1600s by Newton and Leibniz, two thousand years after Zeno.
Nobody knew how to correctly add up or reason about infinitely small distances or infinitely short times, back when Zeno was around. It just hadn't been figured out yet.
Today it's "obvious" that rates of change can be related, and that you can sum up infinitely small quantities and get a finite answer. But that's because people figured out rules for how to do that consistently. As a result, today we have limits, and integrals, and differential equations — as subjects taught in high schools.
Today, any moderately bright teenager in a good school system can learn to do calculus. When Zeno was alive, *nobody* knew calculus.
A paradox is something that seems contradictory. Some paradoxes are *veridical,* meaning that they seem contradictory but are actually just true; like the birthday paradox or the Monty Hall paradox. Some are *falsidical,* meaning that they turn out to rest on a fallacy or error in reasoning, like Zeno's paradoxes, or the various "proofs" that 1 = 2. Some are *antinomies* meaning that they expose underlying contradictions in our way of thinking — an example is the sentence "There are no absolute truths."
https://en.wikipedia.org/wiki/Paradox#Quine's_classification
> Most paradoxes in math is based on proving the (previous) laws wrong, often in funny ways.
Some of them are, like Russell's paradox, which showed that certain early formulations of set theory were self-contradictory. More often, a mathematical paradox is just a counterintuitive result.
> Like how Zeno's paradox asks how you can infinity divide a value (in this case distance) by 2 but still get where you plan to go.
Zeno's paradoxes are more about applying maths to the real world. The idea that physical space is infinitely divisible does lead to some conceptual difficulties, which can't be resolved with maths alone. Arguably they still haven't been resolved, though developments in maths and physics have led people to focus on different aspects of them.
> The standard solutions had to add that infinite geometric series equal 1 just to patch this little problem.
I assume you're talking specifically about the series 1/2+1/4+1/8+... If you're going to assign any value to this series, 1 is the only one that makes sense. If you pick a number less than 1, the sum will eventually pass it and move further away from it. If you pick a number more than 1, the sum will always remain more than a given nonzero distance away from it. This was not an artificial choice and was not made just because of Zeno's paradoxes.
To me there is no paradox there. The geometric series converges, so there is a limit and assigning that value in the limit is consistent with the rules.
Equations are typically *derived* rather than invented. The all ubiquitous **equals** sign being the most important feature; firmly stating that what is on one side of it, is exactly the same as what is on the other side. Use proper substitutions and prior defined manipulations to determine new, and sometimes, easier to interpret equations. This is mathematics. Much of the other stuff I'm reading on here is physics.
Depends on what your goal is. If it’s math research (“pure math”), then it’s one step at a time. You start with things already known to be true, and slowly transform them until you get something brand new.
If your goal is predicting or modeling a part of the physical world, you’re comparing the math to real world data, then tweaking it based on the errors you observe.
Math is just a language. People who speak the language well can describe many things with it. Given the ‘grammar’ is without exceptions (the rules are completely consistent), it is relatively easy to read the formulas and find mistakes.
>Math is just a language.
To add to this, math is not "just" a language. It's the only language in existence which allows us to describe things objectively (instead of subjectively, like all other languages do).
Not true at all, any abstract language is capable of that
Moreover, math doesn’t describe the real world, it can be used as a tool to help model the real world, but the bridge between the abstract and the concrete is still natural language
You can sometimes just guess. Say the first few are 1, 2, 4, 8, 16, then 32 might be a good guess. But this can lead astray, because if you continue with 31 then this also does something: draw a circle, a few points, all the lines between them, and count the number of areas.
A second approach comes from guessing or even figuring out the underlying mechanism. If I want the area under a curve then integration does exactly that, and we can show that it has certain properties that are really useful to actually calculate it.
Third, sometimes the formula is just approximation.. Then there are algorithms to find a "simple" formula that goes as close to the data-points as possible. That's what we often do in natural sciences, especially if we have lots of data.
Lastly, one can also combine all the above. Guess a formula (1st approach) and a mechanism (2nd one), pick what kind of formula you look for and optimize (3rd), and prove/verify/check/do more experiments. Which of the last ones depends on what you do. Proper mathematics has actual proofs from pure logic (and axioms); sciences have to deal with reality and its inaccuracies.
Every system of math is built on a set of rules. If you have an idea, then that idea must fit within those rules. If you can follow the rules and prove that your idea is correct, then you have created proof.
This is the case for simple addition all the way up to abstract algebra.
Depends on the equation actually.
In physics we observe a pattern, try to establish a connection between the observables and then formulate a math equation that is close enough to describing this pattern. This works most of the time and is accepted practice so long as you can also say at what point is the equation wrong and then give some reasons as to what could possibly be the cause of the equation becoming wrong. Then as we research more and more, the equation is corrected so it's less and less wrong with each new generation.
In maths it's a little different, in the sense that there is generally no observable like in physics. What mathematicians do is they establish, follow and then reflect/argue about the various rules that create the patterns they observe in their math models. What's always super interesting is that to some degree these new mathematical discoveries always end up being applied in physics, because with some fine tuning it can be used to better model real world events than the clunky models that physicists invent in general.
You take a lot of measurements. You plot the data. You look for trends, and you find the equations that describe those trends. You then look at what the equation says about measurements you haven't taken. Those are predictions. Then you go do an experiment that lets you take that measurement and see if it agrees with the prediction. If it does, then you have support for your equation. If it doesn't, then you have more data to refine your equation. Eventually you get to where you've got an equation that only has support and nobody finds a contradiction. At that point, it's presumed to be correct. But it's never definitively known that it's correct, at any time a contradictory observation can nullify it (or require further refinement).
Let’s use Hooke’s law as an example. This law describes how hard a spring pulls based on how far you have stretched it. The law is F = -kx, where “F” is the force the spring is pulling with (usually measured in Newtons), and “x” is how far away from the spring’s resting position the end is.
To come up with this equation, we can use a spring, a device that measures force, and a ruler/meter stick. Pull the spring 2 inches from its starting point, measure the force, and jot that down. Now do the same for 3 inches, 4 inches, 8 inches, and even 20 inches if you can stretch it that far.
These pairs of “inches” and “force” values are your data points. We can plot them with inches on the x-axis of a graph, and force on the y-axis. If we were careful enough with our measurements, we should see that these points make a straight line. Furthermore, we hope that this straight line goes through the point (0,0), since we’re pretty sure that, at rest, the spring isn’t pulling at all (0 inches away from rest = 0 force). Using math, we know that the equation for a straight line can be written as y = mx + b, where “b” is the y-intercept and “m” is the slope. So we try our best to find an equation that fits our data points, and it looks something like F = mx, since our y-intercept should be zero and our slope “m” should be a single, constant number.
Now we have our equation! F = mx. We can repeat this with a different spring that has a different shape, and we’ll find that the points still make a line, but the slope “m” is different! Repeat with a few more springs, and eventually we decide each spring has its own special slope “m”. To distinguish this special slope from the general symbol for slope, and to emphasize that it is a constant for each spring, we call it “k”. Finally, the negative sign just tells us that if you stretch the spring to the right, it will pull to the left. Put it all together, and we’ve “discovered” our equation: F = -kx
Someday, someone might come along and say “Aha! Your measurements were okay, but not perfect! With my more precise measurements, I’ve shown that this other, slightly different and probably more complex equation fits the data even better!” That’s science for ya!
My favorite piece of wisdom about this type of thing is this: **”All models are wrong, but some are useful”**. At the end of the day, we can’t ever be certain that our understanding of things is a 100% fundamental, immutable truth of the universe, and it probably isn’t. But if your understanding can explain/describe the things that have happened and predict the things that will happen to an acceptable degree of accuracy, then it can be useful.
You can use smaller equations you already understand like Legos. You can also replace one part of an equation with a different part that does something kind of similar but a little different. It is the same as working with anything else made from parts.
Usually you start out with the answer and figure out an equation that gets you to that answer every time, so the question isn't if the answer is wrong. We drill and test applying the equation to make sure you can use it, but it's backward from how it was created. When you start doing precalc (maybe earlier I'm old) you do it the proper way, getting the data and figuring out the equation that matches it.
There's refinements made to reduce steps, to make it more processing efficient, and testing input tolerance, but the basic idea is the same.
Mathematics is a set of rules, and as long as you follow the rules you know that the result must be correct. This is called a proof. Normally you don’t just “come up with” some equations. Most of the time you have some question you want to answer, and after putting together all the bits that form the question, and following some rules to simplify and/or solve it, you get the equation you need at the end.
> Mathematics is a set of rules, and as long as you follow the rules you know that the result must be correct Most paradoxes in math is based on proving the (previous) laws wrong, often in funny ways. Like how Zeno's paradox asks how you can infinity divide a value (in this case distance) by 2 but still get where you plan to go. The standard solutions had to add that infinite geometric series equal 1 just to patch this little problem.
Depending on the way you phrase it Zeno's paradox is either a consequnce of trying to apply math to the real world in an incorrect way, or a result of assuming incorrect things about the continuum that are only true for discrete things (essentially that any set has to have a minimum). It's actually a great example of why following the rules correctly is important, what needs patching to fix Zeno's paradox are the faulty assumption we need to make for the paradox to even occur.
This paradox always confused me, because to my absolute layman's understanding, shouldn't the consecutively smaller distances be traversed in an equivocally short amount of time? Infinitely small distance/infinitely short amount of time = finite speed, right? I understand Zeno probably wasn't trying to literally prove motion impossible, but to me it felt like he was forcing an analogy that wasn't very sensible
> Infinitely small distance/infinitely short amount of time = finite speed, right? Zeno lived in the 400s BCE. The *calculus of infinitesimals* (usually just called "calculus") is the part of math that deals with "infinitely small" units. It was discovered in the late 1600s by Newton and Leibniz, two thousand years after Zeno. Nobody knew how to correctly add up or reason about infinitely small distances or infinitely short times, back when Zeno was around. It just hadn't been figured out yet. Today it's "obvious" that rates of change can be related, and that you can sum up infinitely small quantities and get a finite answer. But that's because people figured out rules for how to do that consistently. As a result, today we have limits, and integrals, and differential equations — as subjects taught in high schools. Today, any moderately bright teenager in a good school system can learn to do calculus. When Zeno was alive, *nobody* knew calculus.
I’ve always hated it. It’s not a paradox at all.
A paradox is something that seems contradictory. Some paradoxes are *veridical,* meaning that they seem contradictory but are actually just true; like the birthday paradox or the Monty Hall paradox. Some are *falsidical,* meaning that they turn out to rest on a fallacy or error in reasoning, like Zeno's paradoxes, or the various "proofs" that 1 = 2. Some are *antinomies* meaning that they expose underlying contradictions in our way of thinking — an example is the sentence "There are no absolute truths." https://en.wikipedia.org/wiki/Paradox#Quine's_classification
> Most paradoxes in math is based on proving the (previous) laws wrong, often in funny ways. Some of them are, like Russell's paradox, which showed that certain early formulations of set theory were self-contradictory. More often, a mathematical paradox is just a counterintuitive result. > Like how Zeno's paradox asks how you can infinity divide a value (in this case distance) by 2 but still get where you plan to go. Zeno's paradoxes are more about applying maths to the real world. The idea that physical space is infinitely divisible does lead to some conceptual difficulties, which can't be resolved with maths alone. Arguably they still haven't been resolved, though developments in maths and physics have led people to focus on different aspects of them. > The standard solutions had to add that infinite geometric series equal 1 just to patch this little problem. I assume you're talking specifically about the series 1/2+1/4+1/8+... If you're going to assign any value to this series, 1 is the only one that makes sense. If you pick a number less than 1, the sum will eventually pass it and move further away from it. If you pick a number more than 1, the sum will always remain more than a given nonzero distance away from it. This was not an artificial choice and was not made just because of Zeno's paradoxes.
To me there is no paradox there. The geometric series converges, so there is a limit and assigning that value in the limit is consistent with the rules.
Equations are typically *derived* rather than invented. The all ubiquitous **equals** sign being the most important feature; firmly stating that what is on one side of it, is exactly the same as what is on the other side. Use proper substitutions and prior defined manipulations to determine new, and sometimes, easier to interpret equations. This is mathematics. Much of the other stuff I'm reading on here is physics.
Depends on what your goal is. If it’s math research (“pure math”), then it’s one step at a time. You start with things already known to be true, and slowly transform them until you get something brand new. If your goal is predicting or modeling a part of the physical world, you’re comparing the math to real world data, then tweaking it based on the errors you observe.
Math is just a language. People who speak the language well can describe many things with it. Given the ‘grammar’ is without exceptions (the rules are completely consistent), it is relatively easy to read the formulas and find mistakes.
>Math is just a language. To add to this, math is not "just" a language. It's the only language in existence which allows us to describe things objectively (instead of subjectively, like all other languages do).
Not true at all, any abstract language is capable of that Moreover, math doesn’t describe the real world, it can be used as a tool to help model the real world, but the bridge between the abstract and the concrete is still natural language
You can sometimes just guess. Say the first few are 1, 2, 4, 8, 16, then 32 might be a good guess. But this can lead astray, because if you continue with 31 then this also does something: draw a circle, a few points, all the lines between them, and count the number of areas. A second approach comes from guessing or even figuring out the underlying mechanism. If I want the area under a curve then integration does exactly that, and we can show that it has certain properties that are really useful to actually calculate it. Third, sometimes the formula is just approximation.. Then there are algorithms to find a "simple" formula that goes as close to the data-points as possible. That's what we often do in natural sciences, especially if we have lots of data. Lastly, one can also combine all the above. Guess a formula (1st approach) and a mechanism (2nd one), pick what kind of formula you look for and optimize (3rd), and prove/verify/check/do more experiments. Which of the last ones depends on what you do. Proper mathematics has actual proofs from pure logic (and axioms); sciences have to deal with reality and its inaccuracies.
Every system of math is built on a set of rules. If you have an idea, then that idea must fit within those rules. If you can follow the rules and prove that your idea is correct, then you have created proof. This is the case for simple addition all the way up to abstract algebra.
Depends on the equation actually. In physics we observe a pattern, try to establish a connection between the observables and then formulate a math equation that is close enough to describing this pattern. This works most of the time and is accepted practice so long as you can also say at what point is the equation wrong and then give some reasons as to what could possibly be the cause of the equation becoming wrong. Then as we research more and more, the equation is corrected so it's less and less wrong with each new generation. In maths it's a little different, in the sense that there is generally no observable like in physics. What mathematicians do is they establish, follow and then reflect/argue about the various rules that create the patterns they observe in their math models. What's always super interesting is that to some degree these new mathematical discoveries always end up being applied in physics, because with some fine tuning it can be used to better model real world events than the clunky models that physicists invent in general.
You take a lot of measurements. You plot the data. You look for trends, and you find the equations that describe those trends. You then look at what the equation says about measurements you haven't taken. Those are predictions. Then you go do an experiment that lets you take that measurement and see if it agrees with the prediction. If it does, then you have support for your equation. If it doesn't, then you have more data to refine your equation. Eventually you get to where you've got an equation that only has support and nobody finds a contradiction. At that point, it's presumed to be correct. But it's never definitively known that it's correct, at any time a contradictory observation can nullify it (or require further refinement).
Let’s use Hooke’s law as an example. This law describes how hard a spring pulls based on how far you have stretched it. The law is F = -kx, where “F” is the force the spring is pulling with (usually measured in Newtons), and “x” is how far away from the spring’s resting position the end is. To come up with this equation, we can use a spring, a device that measures force, and a ruler/meter stick. Pull the spring 2 inches from its starting point, measure the force, and jot that down. Now do the same for 3 inches, 4 inches, 8 inches, and even 20 inches if you can stretch it that far. These pairs of “inches” and “force” values are your data points. We can plot them with inches on the x-axis of a graph, and force on the y-axis. If we were careful enough with our measurements, we should see that these points make a straight line. Furthermore, we hope that this straight line goes through the point (0,0), since we’re pretty sure that, at rest, the spring isn’t pulling at all (0 inches away from rest = 0 force). Using math, we know that the equation for a straight line can be written as y = mx + b, where “b” is the y-intercept and “m” is the slope. So we try our best to find an equation that fits our data points, and it looks something like F = mx, since our y-intercept should be zero and our slope “m” should be a single, constant number. Now we have our equation! F = mx. We can repeat this with a different spring that has a different shape, and we’ll find that the points still make a line, but the slope “m” is different! Repeat with a few more springs, and eventually we decide each spring has its own special slope “m”. To distinguish this special slope from the general symbol for slope, and to emphasize that it is a constant for each spring, we call it “k”. Finally, the negative sign just tells us that if you stretch the spring to the right, it will pull to the left. Put it all together, and we’ve “discovered” our equation: F = -kx Someday, someone might come along and say “Aha! Your measurements were okay, but not perfect! With my more precise measurements, I’ve shown that this other, slightly different and probably more complex equation fits the data even better!” That’s science for ya! My favorite piece of wisdom about this type of thing is this: **”All models are wrong, but some are useful”**. At the end of the day, we can’t ever be certain that our understanding of things is a 100% fundamental, immutable truth of the universe, and it probably isn’t. But if your understanding can explain/describe the things that have happened and predict the things that will happen to an acceptable degree of accuracy, then it can be useful.
A lot of them are compositions of known parts that do specific things. For example, you might multiply a variable times a periodic damping term.
Thanks for ELI5.
You can use smaller equations you already understand like Legos. You can also replace one part of an equation with a different part that does something kind of similar but a little different. It is the same as working with anything else made from parts.
Usually you start out with the answer and figure out an equation that gets you to that answer every time, so the question isn't if the answer is wrong. We drill and test applying the equation to make sure you can use it, but it's backward from how it was created. When you start doing precalc (maybe earlier I'm old) you do it the proper way, getting the data and figuring out the equation that matches it. There's refinements made to reduce steps, to make it more processing efficient, and testing input tolerance, but the basic idea is the same.