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My 13yr old has leapt forward in math during the pandemic. He's taking discrete math right now but is running into a bit of a wall with proofs. I have a feeling he needs to find reasons why they can be important and useful for him, and doing so will help motivate him to find a way to understand how to master them. I'm dyslexic and mathlexic, proofs are not my forte, so reaching out to you all on Stack for insightful ideas on how to help him make that 'connection'.

What thoughts do you have about why they're important and how they're useful, especially for inventors, engineers, makers, software engineers, and philosophy?

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    $\begingroup$ I think you might get a lot of people who want to challenge the framing of your question, for example many users of this site may want to reply with a sentiment like "The right thing to tell your 13yr old is that most mathematicians do not care if proofs are useful to engineers." The way you phrased the question, you are specifically asking for those answers to not be posted. If you are open to that kind of answer, you could edit your question. Of course you don't have to; I'm just trying to help improve the question if it can be improved. Cheers! Edit: I think the title edit addressed this. $\endgroup$
    – Chris Cunningham
    Jul 30 at 19:38
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    $\begingroup$ @AgentZebra Welcome to Math Educators SE. Note to all: You had posted basically the same question yesterday on the Math SE site at Math Proofs - why are they important and how are they useful & specifically applicable to inventing, engineering, coding, and philosophy, with it now having $4$ answers. Note that cross-posting is generally frowned upon but, if you do it, please at least include links from each post to the other one to help avoid duplicating effort. $\endgroup$ Jul 31 at 0:42
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    $\begingroup$ Would you check your parachute before taking a jump? $\endgroup$
    – copper.hat
    Jul 31 at 21:09
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    $\begingroup$ A quick comment, because I don't have time to write a full answer: while it is true (as others have said) that proofs are used both to verify that something is true, and to explain why it is true, it is also the case that proof can be used to discover if something is true. Often we genuinely don't know whether something is true or not -- and the only way to find out is to write a proof! $\endgroup$
    – mweiss
    Aug 1 at 1:26
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    $\begingroup$ The classic source on "Why is proof important?" is Michael de Villiers 1990 paper, "The Role and Function of Proof in Mathematics", which distinguishes among different roles that proof plays: (1) Proof as a means of verification/conviction; (2) Proof as a means of explanation; (3) Proof as a means of systematisation; (4) Proof as a means of discovery; (5) Proof as a means of communication. Link here: researchgate.net/publication/… $\endgroup$
    – mweiss
    Aug 1 at 1:31

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Proofs are important because proofs are just understanding why something is true. This is what mathematics is all about!

What if all you care about is using the results of mathematics: you don't particularly care about why it works, just how to use it. Should you still care about proofs?

Yes!

I am teaching infinite series in calculus at the moment, so I will use an example from this subject to illustrate some reasons why.

One result which is important in the study of infinite series is that if $a$ and $r$ are two real numbers then:

$$a+ar+ar^2+ar^3+ar^4+\dots = \frac{a}{1-r} \textrm{ if } -1<r<1.$$

We can memorize this result without understanding the proof. However, I will illustrate that understanding the proof gives several practical benefits to the user of mathematics.

  1. Understanding the proof allows us to utilize the intermediate results which lead to the proof.

    Part of the proof of this theorem involves computing the finite sum

    $$ a+ar +ar^2 + ar^3 + \dots + ar^N = a\frac{1-r^{N+1}}{1-r}. $$

    This result is useful in its own right. For instance, this formula is used when calculating savings with recurring regular deposits ($a$) with interest $r$ over a certain number of deposits $N+1$. Note that this intermediate result is applicable even when $r$ is not in the restricted range $-1<r<1$.

  2. Understanding the proof makes it easier to remember the full statement of the theorem, including all of the conditions.

    If you understand the proof of this theorem, then you know the reason why $r$ is restricted to between $-1$ and $1$ is that $\displaystyle\lim_{N \to \infty} r^{N+1} = 0$ if and only if $-1<r<1$. By understanding this reasoning you are building a more robust network of facts in your brain. Placing the fact ($-1<r<1$) in the context of a story (the proof) makes it much easier to remember the fact.

  3. Understanding the proof allows you to generalize the ideas in the proof to new contexts.

    You might need to apply this formula in a new context: for instance, what if $a$ and $r$ are complex numbers? If you only know the theorem in the context of real numbers, and you do not know the proof, then you may not be sure whether you can apply the formula in the context of complex numbers. You would need to consult an external authority to get confirmation before using it. However, if you know the proof you can easily see that "the proof goes through" for complex numbers as well. The only modification needed is that we need $|r|<1$ instead of $-1<r<1$ (which would be meaningless for complex numbers).

  4. Understanding the proof builds a collection of tricks and ideas which can be applied "far away" from the original context.

    There is a cool trick to get the finite sum result I mentioned above:

    Let $S = a+ar +ar^2 + ar^3 + \dots + ar^N$. Then multiply both sides by $r$ to get $rS = ar + ar^2 + ar^3 + \dots +ar^N +ar^{N+1}$. Now $S - rS = a - ar^{N+1}$ since all the other terms cancel. Solving for $S$ yields $S = a \frac{1-r^{N+1}}{1-r}$.

    Learning and understanding this proof gives you an idea which can be used in other contexts. The idea is something like: "Sometimes I have a bunch of things added together in a sequence. If I can shift all of the terms by one, I can subtract to cancel all but a few." More generally the trick might be: "If something is nearly invariant under some group action, then applying the group action and subtracting should 'isolate' the non-invariant part." This idea can be applied in many more situations than just geometric sums.

    For example, when estimating integrals a similar idea comes up when you look at the difference between the left and right Riemann sum for an increasing function. The left and right Riemann sums have all the same terms except for the first and last term. So subtracting gives an easy bound on the error between the integral and the Riemann sum approximation in this case.

    Here we have used the same idea in a completely different context. Without knowing the proof, I wouldn't have this idea in my arsenal.

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    $\begingroup$ "proofs are just understanding why something is true" I disagree. Some proofs confer very little understanding, famously most computer-brute-force proofs, such as the four-color theorem, the solvability of the $3\times3\times3$ Rubik's cube in 20 moves, and the non-existence of a unique-solution $9\times 9$ sudoku with 16 pre-inserted numbers. The core purpose of a proof is to become entirely convinced that something is true. Why is often a nice bonus, but it is not what a proof is. And in general, I am very wary of any question about "why", because that's always a dangerous rabbit hole. $\endgroup$
    – Arthur
    Aug 3 at 9:32
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    $\begingroup$ @Arthur Most mathematicians I know really do care that their proofs psychologically satisfy their need for "understanding". When we find a proof which only demonstrates that something is true, but doesn't give us a good feeling of why, we often persist in searching for more satisfying reasons. $\endgroup$ Aug 3 at 21:05
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    $\begingroup$ It is a noble goal, but ultimately a human one. The goal of mathematicians is to understand why. The goal of mathematics is to determine whether. And that's why computer brute-forced proofs are generally accepted: Because they are just as much proof as the most elegant reasoning. $\endgroup$
    – Arthur
    Aug 3 at 22:09
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    $\begingroup$ Nice add-on to 3.: Somewhen later on, understanding the proof will also guide you to do them same thing for matrices rather than numbers - and voilà, there you have the Neumann series, which is an extremely powerful tool. $\endgroup$ Aug 3 at 23:10
  • $\begingroup$ @arthur I think there is a difference between why proofs are important for an aspiring math student and why they are important with respect to advancing human knowledge. I think this answer well addresses the former question. $\endgroup$
    – Jim Conant
    Aug 14 at 19:23
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I am an engineer. I have not done a mathematical proof since leaving school. Despite that, I believe that proofs are the second most important skill that any student will learn during their entire educational career (second only to basic literacy).

When you do a mathematical proof, you:

  1. Start with a problem, a well-defined set of conditions.
  2. Advance from that starting point one step at a time, with each step representing the application of a theorem, definition, or other principle that is itself already known to be true.
  3. Arrive at a conclusion that either justifies or contradicts whatever it is that you're trying to prove.

This is about far more than math. This process is the basis for rational thought itself. You're not just learning how to prove mathematical theorems, you're learning how to think logically and to reason using facts (not opinion or instinct). You're learning how to take various things that you know to be true and use them to solve problems that are unlike anything you've ever encountered before. You're learning how to use a rigorous thought process to avoid logical fallacies, jumping to conclusions, or being misled by rhetoric.

Most schools no longer teach logic as it's own subject. The same important concepts are generally taught through mathematics, as math is more concrete (thus presumably easier to understand) than approaching logic from the abstract, philosophical side. My school taught proofs in geometry, where problems involved shapes, angles, and other things that could be drawn and easily visualized each step of the way. Proofs were always the hard part of the course. Students never really "got it" until they realized that it's not really about geometry at all. It's about thinking, and the geometry is more of a visual aid used to create a limited workspace with well-defined rules that we can use to practice logical reasoning.

Even though I haven't done a formal mathematical proof in years, I use those skills and go through the same process every day: when I encounter a type of problem that I haven't seen before, when I evaluate what someone said to determine whether it's true or not, when my instincts tell me something is true but I want to make sure that it's actually true, etc.

The subjects taught in school are like the exercises done in a gym. Bench-pressing weights seems pointless, how often do you ever need to lift a heavy stick vertically off of your chest? You do them not because that activity itself is useful, but because it's designed to isolate and strengthen a specific muscle group. After building up strength throughout your body, you'll use those muscle groups together to solve real-world problems. The proofs that you're doing are building the logic portion of your brain. You're learning how to think, a critical life skill. If you know how to read, how to think, and how to learn, then you have the ability to teach yourself anything that you'd ever want to know - and what can be more powerful than that?

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    $\begingroup$ Well put! This gets to the basics! :) $\endgroup$ Aug 1 at 21:50
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    $\begingroup$ @bta Welcome to Math Educators SE. I agree with Paul that you have a great answer. $\endgroup$ Aug 2 at 1:01
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I am an engineer.

Proofs are important to "get" engineering, but are not directly used. I see three aspects of learning proofs as important: Logic, Process, and Ontology.

Logic is the simplest: you need to know how logic works. It's foundational. For example, you will not be able to usefully apply even very basic statistics without being able to apply a clear, logical chain of reasoning. If you try something complex, you are hosed. That example is minor and applicable to every field you mentioned; it is even more foundational for software and philosophy.

Process is harder: as an engineer, in anything but the most trivial tasks, you will need to rapidly learn and manipulate processes and systems. Proofs model this: you take inputs that are given and evaluate and describe the implications. You also Voltron together smaller logic-blobs into larger logic-blobs to reach a defined outcome, which nicely models the engineering design process.

Ontology: engineering is all approximations and uncertainty. You need to be able to bound and contain that uncertainty and to recognize which approximations are close enough. To do that you need to not only know stuff, but also understand how much you can rely on the stuff you know. Proofs are the high end of that; a triangle in a plane always has 180 degrees. Compare that with e.g. Young's Modulus for a material, which is experimentally determined, or the maximum load on something, which is measured or assumed.

None of those say "doing proofs" is critical to engineering, because it's not. By the same token, weight training isn't part of football and singing scales isn't part of vocal performance. Proofs will train your mind for any technical field in a way that is very hard to replicate.


Now all that said, geometric proofs are also independently great for two reasons:

  • You organically learn all the geometry that you need to know anyway (otherwise you would have to memorize it, which is terrible). E.g. I have no memory of what the angle on a regular dodecagon is, but I can figure it out with scratch paper faster than I can look it up (150 degrees; approx. 20 seconds). That's useful for everything from phasors to CAD.
  • They're fun, if you let them be. Geometric proofs are great brain-teasers. If you will enjoy inventing, engineering, making, software engineering, or philosophy, you will very likely enjoy proofs.
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    $\begingroup$ +1 for verbing Voltron $\endgroup$ Aug 2 at 12:54
  • $\begingroup$ Geometric proofs and derivations in kinematics are especially awesome in my mind! $\endgroup$ Aug 2 at 19:05
  • $\begingroup$ +1 for "They're fun, if you let them be" - I had a secondary school midterm where the entire section on geometry was 1 question: it was a jumble of triangles, circles, rhombi, etc, as if a toddler went crazy with their shape toys and threw them on a page. We were told to prove that the length of the base of a right triangle on one side of the page was the same length as the radius of a circle on the other side of the page. It was a blast building the path of truths'n'proofs from one side to the other and vice versa to meet in the middle! $\endgroup$
    – Daevin
    Aug 3 at 16:19
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Along similar lines as the previous answers. Just as a computer program needs to be tested, to make sure it works correctly, a mathematical result needs to be proved to make sure it really does work.

When you are trying to get clearer about relationships between things like numbers and shapes, you might look for a pattern. And if you find a pattern that fits, you might think you've solved the problem. But you don't know that you've solved it until you've given a reason (proof) for that solution to fit that relationship. Perhaps this is too vague. Here's the example I'm thinking of. It starts out like a puzzle.

On a circle, put some points. Connect each point to every other point with straight lines. How many regions can you create for n points?

  • One point can't connect to any others. There is just 1 region in the circle.
  • Two points connect to each other with one line, making 2 regions.
  • Three points connect with 3 lines, making 4 regions.
  • Four points connect with 6 lines, making 8 regions.

Well, there's an obvious pattern here. The number of regions is doubling. And yes, it still doubles again for 5 points. But sadly (or amazingly, depending on your perspective) when we get to 6 points, the pattern breaks down.

We need to know where the pattern comes from to know that the pattern will always work. And finding where it comes from is pretty much the same as proving that the pattern will work.

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  • $\begingroup$ Yes, while it is certainly true that we should start to believe/suspect something based on a bunch of examples (it would be dumb to ignore examples), a bunch of examples is not conclusive "proof" in a colloquial sense. Apart from prank scenarios, there are genuine situations where the smallest examples are misleading, etc. So, again, emphatically, while we should not mistrust heuristics based on examples, we should not consider them definitive. $\endgroup$ Jul 31 at 22:27
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    $\begingroup$ A computer program needs to be tested, but one should also prove (in the mathematical sense) that one's computer programs work correctly! $\endgroup$ Aug 1 at 3:33
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    $\begingroup$ @AlexanderWoo I believe you worked in the software industry for a while, which I never have. Was any of your code proven correct? $\endgroup$ Aug 1 at 15:09
  • $\begingroup$ No (except for trivially small segments). That's why I quit the software industry. $\endgroup$ Aug 1 at 21:30
  • $\begingroup$ @AlexanderWoo Maybe avionics companies are hiring? They do plenty of formally-proven-correct stuff for fly by wire and such. Functional safety systems also could use some of it, although they do rely also a bit on "it's old and it works, therefore it's unlikely to be incorrect". $\endgroup$ Aug 2 at 19:07
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As in @StevenGubkin's answer, indeed, "proofs" are (fairly definitive) explanations why something is true.

I would agree/concede that mathematics is very useful to a variety of people without knowing the proofs. That is, in fact, the "what" of mathematics is already crazily useful...

... which is why sometimes we'd care about the "why". Sometimes, to have more confidence in the "what".

And, yes, I concede/agree that some pictures of "proof" are very stylized, and/or not interesting. E.g., although there is some interest in seeing that we can prove $1+1=2$ from very primitive assumptions (as in Peano and Russell-Whitehead), the added knowledge is not that "finally we know that $1+1=2$", but something about formal logical deduction systems, with that merely as a test case.

Especially when doing somewhat fancier things, we can imagine that we are perhaps stressing the ideas beyond their original operating range. :) So, although we might have many known examples where we can "see" the truth and don't need a dressed-up "proof", in more extreme cases a proof can be critically reassuring.

An example of which I'm fond (which also illustrates the human-fallibility aspect) is to test whether multiplication of large integers is commutative. Yes, we "know" that it is, and can also "prove" this from Peano or other foundations. But, in contrast, if we just "test", by hand-multiplying two random-ish 20-digit decimal expansions, most likely we will seem to find that $a\cdot b\not=b\cdot a$. :) Duh, because out of the 400 single-digit multiplications and lots of additions, the chances are high that we'll make a mistake. :) One of the aspects, too, is the brittleness of something like this computation. It's not self-correcting. In contrast, a good proof style (probably not the toooo stylized/caricatured ones) is that it is robust, in the sense that we can sensibly review it, and also that the pieces fit together in a mutually-correcting way (in human psychological/linguistic terms).

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    $\begingroup$ Strong +1 for: the added knowledge is not that "finally we know that $1+1=2$", but something about formal logical deduction systems, with that merely as a test case. This captures what I was thinking a couple of weeks ago when I wrote some (probably too many) comments to the MSE question Is reading less rigorous (and partially historical/contextual) math “textbooks” a waste of time?. I'm pretty sure (now) the OP knows the difference, but I often read things that lead me to think someone is conflating logical rigor with logical formalism. $\endgroup$ Jul 31 at 18:30
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    $\begingroup$ I'd say that most modern, computer aided, systems like Russell-Whitehead, like Metamath and Coq, are in theory trying to provide trust to results what we aren't sure is true. The Flyspec project took a proof of Kepler's Conjecture that reviewers were 99% sure was true and made a computer proof of it. Metamath also effectively provides an book of mathematical proofs at a consistent careful level, meaning if you find you don't understand a proof, the Metamath proof, if there is one, will provide all the details. $\endgroup$
    – prosfilaes
    Jul 31 at 19:49
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You need a proof to know that what you've observed is true beyond the cases where you observed it.

How do you know that the sum of two odd numbers is even? Most kids say it's obvious after they've seem lots of cases - where "lots" means a few with really small odd numbers. The proof isn't hard - once you have good definitions of "even" and "odd".

How do you know there are infinitely many primes? There are - Euclid proved it. How do you know there are infinitely many pairs of primes two apart, like 5 and 7 or 101 and 103. You don't. No one does. If you can prove it or prove it is false you will be famous (in a small circle).

How do you know that every even number is the sum of two primes? That's known to be true for all the evens up to some huge number. You can find the current bound reading about Goldbach's conjecture. But no one knows whether it's true forever. If you can prove it (or find a counterexample) you'll be famous .

I hope that in your son's studies he is asked questions like "prove or disprove X" where X is a statement interesting enough to play with. Sometimes it will be true, sometimes not. Finding out which and writing a convincing argument for your conclusion (that is, a proof) is a good way to learn why proofs are necessary and how to write them.

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  • $\begingroup$ Yes - if you are not just listening to what's known, but you're trying to find out something new, proofs are a main approach. $\endgroup$
    – usul
    Aug 1 at 0:43
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    $\begingroup$ @usul I don't think proofs are a main approach to finding out something new. You start by playing with examples, honing your intuition. Then the search for a proof is the next step. $\endgroup$ Aug 1 at 0:46
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    $\begingroup$ By "find out", I didn't mean guess or conjecture, but actually decide. $\endgroup$
    – usul
    Aug 1 at 0:47
  • $\begingroup$ Very relevant answer. Learning techniques how to prove or disprove a claim (made by someone else or by yourself), is one of the most important skills in an educated society. $\endgroup$ Aug 2 at 15:29
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Fundamentally, proofs draw a conclusion from some premises. This is how you apply them in the rest of life. If you observe (or assume) the premises are true in real life, the conclusion is true as well. You don't have to go through the effort of verifying the arduous chain from premises to conclusions. Once you have verified the proof, you can just use it!

I find the word "verify" to be very important in the above paragraph. One of the things that makes proofs so gosh darn powerful is that you can sit down and work through their logic to determine if the conclusion really follows from the premises. If your son is having trouble writing proofs this is to be expected: it is far harder to write a proof than to verify it.

And, indeed, that is the point. A proof succinctly captures something enormously difficult to think through and puts it in a form that is easier to verify.

You mention software engineering. I recently had to put together a piece of software that needed to demonstrate that some condition was true before proceeding. I had access to some data about the present conditions, and needed to process it to see if it "entailed" the condition I was looking for. With the data I had available, this called for a particularly difficult class of algorithms. Fortunately, I was not the first person to want these algorithms. I could go out on the internet and find papers about how to do it, written by multiple PhDs.

But could I trust it? Sure, I could code it up, but how could I be certain that it would handle all of my scenarios? The answer was that the papers I read contained proofs. I could go through them, line by line, and see why the algorithms I was looking at were "complete." And this is really interesting, because I don't have a PhD. In fact, my forte isn't even in the field these algorithms come from. There is no way I could have worked my way through these on my own, but the authors had done the hard part. They walked me through the steps and the leaps of inspiration I wouldn't have thought of, proving the algorithm did what they said it did.

And then I built on it, with my own proofs. They were nowhere near as difficult as the proofs I found for the underlying math, but I could prove that, assuming their algorithms did what they say they did, mine would do what I said it did as well. And this is something that makes proofs very powerful. When you're just learning, you're only thinking about one proof at a time. But they chain. We build upon the lessons from others. Proofs let us stand on the shoulders of giants and peer just a little further.

This transitive property is king. It is what makes mathematics as powerful as it is. It lets you build up giant castles of knowledge, built on the bedrock of simpler math that you understand. It also lets you explore that simpler math, and see how deep the bedrock truly goes -- how high of a castle do you dare build on it.

And the process of learning to write proofs is useful too. Proofs define a criteria for "this is true," a bar which one must achieve in order to claim that something is actually true. And it provides tools to learn how to demonstrate such things. These tools apply outside of mathematics as well. When I am at a design review, defending my software, I don't present mathematically accurate proofs, but I do go through the same process of helping the reviewers walk through my logic and come to an agreement that it is sound. The difference is that the rigorous proofs of math could prove that what I say is true, while I am otherwise forced to hold reviews to demonstrate that it is true.

As a parting philosophical topic, mathematics often explores really tricky cases. Consider Zeno's paradox: someone trying to run across a field must first run half way across it. And they must first run halfway across that halfway, and so on. This is an infinite number of steps that have to occur before you move. Thus, Zeno concludes nobody moves at all!

Working with these infinities are tricky. I'd argue it took over a millennium before we had precise enough wordings to grapple with them. And it required proofs to avoid paradoxes like the ones Zeno put forth. If your son is interested in reading about math, "Gödel, Escher, Bach: an Eternal Golden Braid" by Douglas Hofstadter is a wonderfully accessible work covering such topics and the search for truth. I like it because it not only demonstrates the power behind such formal thinking, but also shows the process of getting there. I find it hard to separate proofs from mathematical history. Seeing how people formed incomplete ideas into cohesive proofs to share with others is the core of mathematics.

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    $\begingroup$ it is far harder to write a proof than to verify it --- This is usually the case, but counterexamples are quickly found if you've graded enough student papers or tried to carefully follow every step in certain published papers. In fact, my difficulty in following all the cluttered details in the proof of a paper my Ph.D. adviser gave me to read (he correctly thought the result easily could be generalized in a couple of ways) led me to start all over and come up with a more uniform way of proving such results, which I then applied to several related results. (continued) $\endgroup$ Aug 1 at 18:22
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    $\begingroup$ Of course, my adviser -- despite eventually liking my approach "in theory" -- had difficulty in following all the tweaks and refinements I introduced in order to obtain maximal applicability of the method. Sometimes when the proof is a mass of cluttered details, it's easier to create it than to follow a similar type proof that someone else wrote. I imagine this is similar to how it's easier to write overly complicated sentences (without realizing how overly complicated they are) than it is to parse someone else's overly complicated sentences. $\endgroup$ Aug 1 at 18:22
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Proofs are the whole point of mathematics. They are how we verify and explain that we know things instead of merely guess at them. When I personally teach discrete mathematics, the first-day opening that I use to address this issue is this:

Consider a function defined on natural numbers $n$: $f(n) = n^2 - n + 11$. Are all outputs of this function prime?

Inevitably in my classes (at a community college in CUNY), the students quickly and unanimously convince themselves of one answer at the end of the first day, at which point I stop and suggest further things to think about (including the sketch of a program they could write, as this is at least second semester in a CS program). They next day we return and find out they were all incorrect, despite the certainty they had on day one. The point being: proofs are important in this class because they let us distinguish between when we really know and when we only think we know.

Depending on the OP child's maturity (and also the OP), they might consider trying this exercise together.

Finally, on the issue of use for engineers, here's a quote that always sticks with me. From Stein/Barcellos, Calculus and Analytic Geometry, 5E, "To the Instructor", p. xxii (1992):

At the Tulane conference on "Lean and Lively Calculus" in 1986 we heard the engineers say, "Teach the concepts. We'll take care of the applications." Steve Whitaker, in the engineering department at Davis, advised us, "Emphasize proofs, because the ideas that go into the proofs are often the ideas that go into the applications." Oddly, mathematicians suggest that we emphasize applications, and the applied people suggest that we emphasize concepts. We have tried to strike a reasonable balance that gives the instructor flexibility to move in either direction.

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    $\begingroup$ P.S. Horribly, some of my students still hold to the wrong answer even after a counterexample is supplied to them. Eek! $\endgroup$ Aug 1 at 14:46
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    $\begingroup$ I was thinking that if it weren't for the difficulty in checking the arithmetic details, then looking at the roots of the expanded form of $(x-1)(x-2)\cdots(x-10)$ is similar: $x=1$ makes it zero, $x=2$ makes it zero, $\dots,$ $x=10$ makes it zero, $\ldots$ After thinking about that, the following variation of a well-known math joke occurred to me: Given real numbers $a, \, b, \, \ldots, \, z,$ how many turning points does the graph of $f(x) = (x-a)(x-b)\cdots(x-z)$ have? (Intended answer: NONE) $\endgroup$ Aug 1 at 16:08
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I strongly disagree with "proofs are understanding why something is true". The statement holds only in a minority.

I write because I've been told time and again by professors to be content with proofs as intuitions or explanations on why something is true. Stokes' Theorem, electromagnetic fields, derivatives, Fourier transform, etc - I'm pointed to heavy-duty math and told that's the best there is. This has been very wrong every single time.

For advanced topics, proofs are generally useful as explanations only if someone already has a significant background in the field and its underlying mathematical field. For beginner topics, a beginner can't comprehend actual "proofs", only derivations - these make up most of the minority I refer to. Indeed the derivation of the quadratic formula is quite useful, and knowing how to derive is very important, else we're blind-faithing, which is anti-science.

My background is electrical engineering, signal processing, and machine learning via calculus and linear algebra. This post is a response to other posts in the general context rather than OP's, so it's not necessarily to invalidate other posts. YMMV in other fields and maths.

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    $\begingroup$ Interesting take. I agree that for things like Stokes' theorem most proofs aren't explanatory. To understand the theorem you need to reason intuitively with infinitesimals. Real rigor is tedious until/unless you have mastered some advanced differential geometry. But (as you acknowledge) for the OPs context (and for much of discrete mathematics) a good proof is really a good explanation. $\endgroup$ Aug 1 at 0:07
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    $\begingroup$ As an applied, but mathematician, I disagree with you. But see my answer below, there is so much more than just using maths as a tool. $\endgroup$ Aug 1 at 0:21
  • $\begingroup$ @OlegLobachev With how you're treating proofs I of course agree. I more refer to "to understand X just check out these bunch of convoluted formulas that derive it". $\endgroup$ Aug 3 at 4:52
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Proof techniques are problem solving tools just as much as wrenches and ratchets are.

Consider proof by contradiction.

The other day when I was fasting I got suddenly concerned I might've drunk soda pop while at home. So I assumed I did. Since I drank soda, there must be at least one open can whether partly full or empty. Then I checked for open soda cans in the fridge, countertop and trash bins(all soda in the house is of canned variety). There were none. Thus the assumption was wrong and therefore I didn't drink any soda.

One time I dosed off with a laptop in my hands and when I woke up a program was open with some data exposed in clear text. I got scared for a bit that I might've accidentally erased some data. I read the text beginning to end and everything held up logically. There were no obvious holes that would indicate data loss. Then I assumed data loss and supposed I found the lost data. Adding that data back would, then, break the logical flow of the existing data. Contradiction. Therefore no data loss occurred.

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There are a lot good answers here, but I'd like to supplement with some other benefits.

First, besides being an exercise in reasoning, mathematical proofs are exercises in language use particularly in regards to being precise in syntax. As someone who programs, I can tell you that attention to the details of a syntax are tremendously important. One misplaced or wrong character in a program of a million lines of code can cause the system to fail to build, or even worse, to build and allow subtle errors to occur. Good mathematical proofs contain healthy doses of syntax. Finding compile-time errors is pretty easy. The system barfs on you. Finding subtle run-time errors is a job skill.

Second, proofs are actually an object of study themselves by mathematicians in the discipline of mathematical logic. Both proof theory and model theory explores proofs in the abstract, and those both go to mathematical foundations in the philosophy of math.

Lastly, the foundations of mathematics answers the question "What is mathematics?"; however, mathematics and logics are a foray into the exploration of metaphysics and philosophical questions. Take for instance, Is the brain a computer? (PhilSE) Questions like these delve into deep notions like theories of truth, the nature of justification, the nature of poor reasoning, and so on. Simple math proofs are a simple introduction into philosophical discourse, and have been at the heart of very famous philosophical clashes and events like Frege's invention of formal systems and his resistance to psychologism which still are argued today.

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    $\begingroup$ In the context of mathematical proofs, syntax, reasoning and communication are indeed intertwined. $\endgroup$
    – ryang
    Aug 3 at 9:10
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I agree with some answers, but will phrase it a bit differently.

The theorems are "what". The proofs are "how" and "why".

But not even this is the full beauty of maths. You start with some formal definitions, the axioms. Basically, you let things be like this, assume that in a new, empty, from-scratch world things go like that. From those axioms you can conclude some further statements, the theorems. You can prove them, basing on the properties of the prior facts. (As I often tend to say, the art of a mathematical proof is to obtain a stronger statement by chaining weaker forms of already known statements.) Then, repeat, repeat, repeat. You are building a beautiful building from reasoning on top of the axioms you chose.

And now, the crux. There is not a single set of axioms, you can use multiple. Of course, many would use the already existing sets, as it suffices in that field. Things like the euclidean axioms of geometry on a plane. Or ZFC. Or some kind of a logic. Or category theory. But you can totally switch to a different set of axioms if you need to. Or even a build new one, if you need to.

Next, about proofs. There are proofs that basically show, that if the theorem in question does not hold, then the world is broken. Those are the contradiction proofs. But there are also proofs, that are constructive. Basically, a theorem merely states "foobar exists". A constructive proof would show you, how to build a foobar (whatever it is).

Proofs can be tedious and exhausting, which is why they are often excluded or only glossed about in the typical school setting. I daresay, if a child is interested enough in the subject, then using the typical childish curiosity, "and why is that?", is a possible way to tackle proofs. "It is because of this, but you might grow tired from all the details."

Next issue, the proofs are typically written down to be watertight. To convey all the needed information in all the detail. (Later, they might miss out some steps, but leave enough to connect the dots, more on it below.) One thing, which is in my opinion just as essential as a correct proof, is the proof idea. A rather short and a more approachable summary of how the proof works. ("Square root of two is irrational, because if we'd assume it is not, then it's a reduced fraction. But we would be able to reduce it once more, which is a contradiction")

Now, for connecting dots. During my studies quite some time ago, I stumbled upon a formal definition of what is "can easily be seen", "is trivial" and the sort. It basically means, the intended reader should be able to figure it out in 15 minutes with a paper and a pencil. Which is a lot for an unprepared mind. Basically, you'd need to build up some kind of a frustration tolerance in a young mathematician, ensure they don't think of themselves as dumb and unworthy if they cannot figure a step in the proof, if they cannot connect the dots immediately. Asking around helps. Oral explanations tend to go in the direction of the "proof idea" and hence help with the understanding a lot. (Which is why lectures are important.)

To summarize: Theorems are the "what", proofs are the "why". One of the best elementary examples of this way of thinking is classical plane geometry, the things Ancient Greeks knew. Geometry is much closer to "real mathematics" than any other pre-university math-related subject. It takes some skill to read the proofs, to extract the "idea" from it. Aspiring mathematicians need to be wary of trying to gasp everything in this very instant and without help. Some kind of explanations on your side or video lectures might also be a great help.

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Example: Board Game

You are playing a game with a single stone on a hexagonal board. You have cards indicating one of the six directions on the board, which move the stone in the respective direction when played.

Now, your stone is starting in the centre of the board and you have six cards, one for each direction. You have to play each card exactly once, but you can choose the order.

Illustration of the board game

On which spaces can your stone end?

This task it not challenging if you know the basics of ℝ² vector arithmetics, in particular commutativity, which tells you that the order of cards does not matter and the stone always returns to the centre, because the hand of cards is composed entirely of pairs of inverses. If anything, the difficulty is recognising that you can apply this, i.e., applying the right abstraction (vectors) to the problem. Still, even if you have all of mathematics at your disposal to throw at this problem, you will likely perform a little proof to convince yourself of the solution, even if you may not recognise it as such or just think of it as a calculation.

Now, suppose you try to prove the solution without knowing (or applying) vectors. In that case, you will inevitably gain some insights on the mechanics of this game. Most importantly, you will likely discover that the order of cards does not matter – which can be useful in other scenarios (different cards) but also translates to other practical problems. To prove this you may, e.g., look at the sides of rhombuses and learn or reinforce some geometry in the process, or introduce tuples of numbers to describe positions and thus discover vectors.

Sidenote: A student may claim that it’s obvious that the order does not matter. You can demonstrate that this is not so obvious by placing different counters on the spaces that are collected when the stone is moved.

What does this demonstrate?

  • If you are working with mathematics or similar, you are likely to prove some small thing frequently, even though you may have internalised many (proof) patterns to an extent that you don’t even recognise them as such.

  • Proving things grants you a deeper understanding of the problem at hand, allowing you to better solve similar problems.

  • Trying to prove things challenges you to find patterns, abstractions, and similar that are useful beyond the specific problem you are trying to solve. Formulating these is often the main accomplishment of a proof, while “the proof itself” can be a mere technical exercise or calculation.

  • Proving can often be more easy than empiricism or brute force.

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A proof is a precise justification given precise premises that a precise statement is true.

If you don't have a proof or you don't reasonably believe that someone has seen a proof, you don't have a reasonable reason to believe it. Or a reasonable reason--a reason--reasoning--to believe it. A proof is the reasoning that something is so. That is "why they're important and how they're useful".

We mostly get things done by accepting that certain things have been proved in mathematics or reasonably shown likely in engineering and everyday life--often aided by mathematics.

However,learning how to understand & generate proofs of simple precise statements helps us to learn to reason in less precise contexts.

Other answers here are wrong & misleading to say that a proof tells us or shows us "why" something is true. It does not. It shows that it is true. Having a proof is why we are justified in saying it is true. There is no "why" in mathematics; given some premises, other things follow. The only way a proof can reasonably though sloppily be said to say "why" is in the trivial sense that we are justified in saying it is true "because" a proof exists as demonstrated by a proof having been given or proved to exist--the conclusion is true "because of" the premises.

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    $\begingroup$ Can you give an example of how you use the word "why"? I cannot understand how "a reasonable reason to believe [something]" doesn't answer the question "Why is [something] true?". $\endgroup$ Aug 1 at 11:43
  • $\begingroup$ I said the only reasonable use of why in math is trivial & I already gave examples of such claims in my post--'it is true "because" a proof exists' & 'the conclusion is true "because of" the premises'. I don't understand how you think "a reasonable reason to believe [something]" is "Why [something] is true". You seem to agree that having a proof is a reason to believe--but that is true of any proven claim--hence, "trivial". I suggest you pick a proof and justify that it says "why" something is true in any more specific sense. The proof merely shows that it is true & a way to show it. $\endgroup$
    – philipxy
    Aug 1 at 11:52
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    $\begingroup$ My experience as a mathematician is in conflict with your claim. It is very common for mathematicians to ask each other "why" something is true. The answer is often a proof, or the major idea in a proof. Basic example: "I noticed that the sum of the digits of any number divisible by 3 is also divisible by 3. Do you know why that is?". Answer: "compute the number mod 3, and realize that every power of 10 is congruent to 1 mod 3". Very common kind of dialogue in mathematics. Is this usage peculiar? I am asking for ANY example of how "why" is supposed to be used ... $\endgroup$ Aug 1 at 12:52
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    $\begingroup$ My point is if you are teaching someone re proofs & you find yourself wanting to say that something in the proof is "why" the conclusion is so you are really trying to say something about the structure of the proof, and not about the proof actually saying "why" the conclusion is true. (A different proof would have different "why"s. So, why exactly are there unlimited whys? Rhetorical.) I don't wish to discuss Archimedean nth causes & other takes on normal/everyday "why". I'm done now. Thanks. $\endgroup$
    – philipxy
    Aug 1 at 13:20
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    $\begingroup$ I do understand that some people do not believe that any sort of logical implication can be any sort of causality... but in my own practice as a mathematician I imagine that what I write is a descriptive narrative about phenomena... which have causes, etc. $\endgroup$ Aug 4 at 20:49
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As other answers have already well-addressed the question of, "why proving something is useful/important ?", I'll address the related but different question of what personal benefits do I get from the practice of proving/attempting to prove something ?

Firstly, trying to understand a problem statement (of something that is wanted to be proved) is a task in and of itself- and this exercises your mathematical comprehension (related but different than English/language comprehension because maths comprehension is usually more abstract than people-focused).

Usually there are many different proofs for a given maths problem. Comprehension is also exercised when you try to understand other peoples' proofs that maybe you did not come up with yourself.

Secondly, I think proofs are important because they improve your problem-solving skills through creative thinking and abstract imagination, especially when the proof is difficult - it requires you to think outside of the box.

Thirdly, proofs can improve your communication skills because often times it is easy to see the idea behind a proof but it is difficult to actually write down the proof concretely. Furthermore, coming up with your own proof which is different to other proofs sometimes gives you the opportunity to help those who did not understand the other proofs.

I'm sure there are many more you could add to this list...

I have no doubt my answer could be improved, but I think the essence of what I am saying is important: that many important skills are exercised and improved when one does (mathematical) problem-solving.

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Philosopher here.

Mathematical proofs are formulated in a logical language (usually some sort of predicate logic, to a lesser extend propositional logic). Remember that all the tools like proof by contradiction, modus ponens, etc, are logical tools rather than mathematical tools.

Logic provides a framework that has very powerful claims. Let's name a few of them:

  • consistency: There must not be any false formula within a given theory, otherwise the theory could produce any random contradiction. Do not confuse this with the framing of physical theories.
  • soundness: A valid logical argument whose premises are all true has a true conclusion. The other way around: A logical argument of which a contradiction can be derived shows that one of the premises is false. This observation is especially useful to check if something you assume is really true.
  • axioms: Axioms are well-chosen contingent formulae (formulae whose truth values vary depending on interpretation) which are considered to be true within a given theory. All other true formulae within this theory can be derived from them, and no false formulae can be derived from them. The following observation can be made: As soon as you can derive your assumption from axioms, it must be true.

For a 13yo (or any school-level), mathematics is usually a single theory.

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    $\begingroup$ As a philosopher you would know that there are different schools of philosophy of math : platonist, logicist, formalist, intuitionist. And you are essentially selling the logicist/formalist stance (which is admittedly widespread post Cantor&Hilbert) You could do better however by elaborating how in platonist, intuitionist schools the notion of proof differs. And more importantly, its centrality is not what it is in the formalist logicist schools $\endgroup$
    – Rusi
    Aug 3 at 12:06
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Let me give an answer which frames proof as a step towards programmability.

Technically, there are proofs which are not constructive. Typically such proofs rest on some application of proof by contradiction. I don't mean to speak to such proofs. Instead, I'll focus my answer on proofs which are constructive in the following sense:

A constructive proof gives a set of instructions on how to solve a given problem for completely arbitrary inputs. Or, at least for inputs which are suitably constrained.

This sort of proof is not directly programmable without a lot of work in practice. For example, in Linear Algebra you might have a list of items where your proof selects from the list a particular member with a desired attribute. Then, often the argument goes forward by relabeling that desired choice as the first in the newly formed list. This sort of renumeration is easy enough to write in the parlance of typical theoretical mathematics, but, how do you actually achieve that for a given problem using a programming language on a given computer ? I worked for a year outside academics and I learned there the difference between proving and applying. I worked with people who were able to convert my index relabeling arguments by clever filtering and data sorting. Matlab has a deep toolset for those who are schooled in these methods.

I argue:

Direct proof is easier than application.

To apply a direct proof to real world data you have to choose data types and prepare for differently sized sets of data ideally. The rough sketch of how to set-up such a program is given by a direct formal proof. In short,

Proof is a first step towards building a general program.

A proof is akin to a flow-chart which can be understood by an audience of your mathematical peers.

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    $\begingroup$ Yes, indeed, this is one important application of "proof". Part of the point is the intended audience... usually not absolute beginners, so no wonder if absolute beginners may fail to understand what's going on. :) I've certainly seen naive young people doing very naive things in coding... not realizing that they were relying on (false) "facts", etc. :) $\endgroup$ Aug 3 at 21:00
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Learning how to come up with a math proof is good for a kid. It's a problem-solving activity.

Students believe what they are told by their teachers, especially in math class, and with good reason because math teachers rarely teach an untrue theorem, and proofs often are used instead of good explanations. The proofs are looked at, and then forgotten, with nothing gained.

Sure, students probably should be more skeptical, but then again, maybe that skepticism should not be about the math the teacher is teaching them. But teaching students math proofs is not an efficient way to make them skeptical. Showing students proofs that look valid and then showing them how they've been duped might work to make the students skeptical, and even create some genuine interest in proofs.

You could try showing your son this video by 3Blue1Brown called "How to lie with visual proofs" https://www.youtube.com/watch?v=VYQVlVoWoPY and another good idea might be "How to Lie with Statistics" by Darrell Huff. Probably everyone should read the latter.

You say, "especially for inventors, engineers, makers, software engineers, and philosophy" and to that I'd say there's some use in some parts of philosophy, but not among the other groups you mention. These people, like most nonmathematicians, can greatly benefit from knowing how to use math, but rarely find it useful to be able to prove that a piece of math is true. If the math community says it's been proved, they can be safely believed, I think. All those groups you mentioned, with the exception of some types of philosophers, need to master, besides math, the art of testing, and the art of using and making computer simulations. On the other hand, if the student uses creativity to come up a proof, the increase in general creativity that could be expected to result would be useful.

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  • $\begingroup$ One helluva sensible argument +1! Allow me to sharpen it some. Ramanujan was one of the greatest mathematicians of the 20th century (some like to say of all time; but that's by counterfactual extrapolation from an early death. V great is great enough for our purposes...) And he couldn't care less about proofs. If you'd ask him he'd say his chosen deity dictated the mind bending results to him. Clearly my (meta!) proof is non constructive but it IS existential ie we don't know how to produce more Ramanujans but his existence and methods show our proof-focussed approach to be myopic $\endgroup$
    – Rusi
    Aug 3 at 11:31
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    $\begingroup$ @Rusi Very interesting what you said about Ramanujan. This New Scientist article says the same thing: newscientist.com/article/… $\endgroup$ Aug 4 at 20:14
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    $\begingroup$ I've +1ed this (again!) Thought I'd explain! At 1st version I upvoted for the "overrated". When you toned it down to "only for kids" I somewhat reluctantly removed my upvote. Your most recent thorough change gets back my upvote. Not for the silly reason that you agree with me but for the deeper reason that you are evidently one of those odd individuals who's not incapable of changing his mind $\endgroup$
    – Rusi
    Aug 5 at 3:54
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    $\begingroup$ Yeah SE is a bedlam marketplace. The loudness of the ignorant can be deafening. Yet, one can always learn from the exchanges... A more secular examole than Ramanujan would be "Music : scales :: mathematics : proofs" Most musicians need scales but some of the greatest (eg Martha Argerich) despise them $\endgroup$
    – Rusi
    Aug 7 at 3:20
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    $\begingroup$ @Rusi "Wise men learn more from fools than fools from wise men." -Attributed to Cato the Elder. $\endgroup$ Aug 7 at 10:05

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