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I know someone I really like, but sadly, that person has absolutely no experience in math or mathematical thinking above third grade mathematics (+, - are fine, but division already makes problems). This person is an adult and not stupid, but he never had any opportunity to get more educated and it worked for all his life like this.

Now, he asked me why you'd need proofs for anything ever, since it's enough to see that an equation works after testing it for a few values.

What very practical example could I use where an equation works for like the first 100 numbers in $\mathbb{N}$ or something like that, but fails afterwards? Something that doesn't need a long introduction into equations (since this is above his head), except for things like "? + 5 = 10, what is ?" ($x$'s needlessly complicate this since it "looks too mathy").

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    $\begingroup$ There are many interesting examples of patterns that break or eventually fail (math.stackexchange.com/questions/111440/…). I think it is important to also note that math builds on itself, and we cannot build upon a foundation of theorems that are true sometimes. $\endgroup$ – Carser Jan 15 at 12:15
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    $\begingroup$ Previously: math.stackexchange.com/questions/514/… $\endgroup$ – user3067860 Jan 15 at 18:34
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    $\begingroup$ @Dave L Renfro - Without the "x"s this just looks like a random set of numbers. The next number could be anything. Guessing the next number of an ascending series gives you an infinite choice unless you already know the rule. This person struggles with anything beyond addition and subtraction. You think they can multiply out brackets? $\endgroup$ – chasly - supports Monica Jan 16 at 12:32
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    $\begingroup$ It strikes me as incompatible to (a) not understand division, but (b) to formulate the idea of testing an equation with a few values. The latter is a fairly sophisticated thought. Maybe by "division already makes problems" you mean he can't calculate divisions. $\endgroup$ – Joseph O'Rourke Jan 16 at 15:12
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    $\begingroup$ Everything is until a "rule" of some kind is found. --- Incidentally, this is not just a math issue. The OP said the person is an adult and is otherwise intelligent, so they should understand things like "innocent until proven guilty", "known to have committed the crime but we can't prove it", and general philosophical epistemology issues (even if they don't know the technical term "epistemology", at some point in their life they surely have been involved in discussions with someone (or had personal thoughts) in which the underlying concepts are contemplated). $\endgroup$ – Dave L Renfro Jan 16 at 16:50

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More fun than equations are patterns that seem to hold. Put a dot on a circle, connect it to all the other dots (none yet), there is 1 region. Second dot connects to first, two regions. Third dot connects to first two, there are now 4 regions. It sure looks like the regions are doubling. In fact, they are not. It's a fun problem, and takes you by surprise. After you find a number pattern that works better, you will want to make sure it always works (ie prove it).

Example: Put five dots on a circle and connect them all through straight lines. The number of regions is 16.

Example

My paper on this: https://scholarship.claremont.edu/jhm/vol1/iss2/7/ (download is free)

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    $\begingroup$ Most of the answers here, I find amusing - they are so obviously written by mathematicians who have no concept of some people's lack of knowledge of the subject.. If the person has trouble with division, they won't understand prime numbers, algebra, geometrical constructions, etc. etc. At least this answer relies solely on being able to count and multiply by 2. $\endgroup$ – chasly - supports Monica Jan 16 at 12:01
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    $\begingroup$ @chasly-supportsMonica Well your contention in these comments basically boils down to "no math allowed", in which case you should be praising the non-mathematical answers that make analogies to real-world non-math situations, and not the "only multiplies by 2" ones. $\endgroup$ – zibadawa timmy Jan 17 at 2:54
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    $\begingroup$ I think that this answer might be improved by a picture. It took me a little bit to understand what your example was actually talking about. $\endgroup$ – nick012000 Jan 17 at 7:32
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    $\begingroup$ +1 because my immediate thought on seeing this question was to post this exact example, but you saved me the effort. $\endgroup$ – J.G. Jan 17 at 12:51
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    $\begingroup$ @nick012000 I added an image for illustration (pending approval). $\endgroup$ – Vincent Jan 17 at 13:53
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Most answers describe proof as an improved correctness checking tool. But, I don't think this gets at the core of the issue. The goal of math isn't to check off theorems, but to understand them and consequently nature (broadest possible meaning of the word).

Proofs are a way to describe insights into a subject in a clear manner. Impossibility proofs tell you "why" something can't work. Constructive proofs show you "how" to make the thing you want. In general, good proofs are those with explanatory power and insight, they expose what's going on. Consequently, writing a proof forces you to understand the problem, making it a fantastic learning process.

The fact that mathematicians continuously revisit the same subjects and theorems (lines, circles, etc) to develop new ways of looking at them supports this view.

Even simple problems require a lot of understanding. How many subsets are there of a given set? A natural thing to try is writing them for a small set like $\{1, 2, 3\}$. Notice that for the number 1, there are subsets that contain it, and subsets that don't. Those that don't include 1, are the subsets of $\{2, 3\}$, those that contain 1 are not. So if we remove one element $x$ from a set $A$, and find $P(A - \{ x \})$, all of those sets are still subsets of A and none contain x. Adding x into each one creates all the subsets to complete P(A). Each additional element in a set thus double the number of subsets. After checking the empty set (base case), you can conclude $|P(A)| = 2^{|A|}$.

Writing out a table would tell you the same answer, but it wouldn't give the same insight into the subject (or certainty). As a a bonus, once we learn why it's $2^{|A|}$ we also know how to construct subsets with an algorithm.

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    $\begingroup$ "Even simple problems require a lot of understanding." Yes they do, and talking to someone who doesn't even understand division, about sets and subsets and using the notation you have, will simply go over their heads. $\endgroup$ – chasly - supports Monica Jan 16 at 11:57
  • $\begingroup$ @chasly-supportsMonica I absolutely agree. My intent was to share an example I personally thought was insightful. You would need to tailor questions and presentation for the student. $\endgroup$ – Justin Meiners Jan 16 at 17:03
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    $\begingroup$ @chasly-supportsMonica this example is very easy to explain to someone in a way that avoids the notation: grab some small, non-fungible objects, and a sheet of paper. Objects on the paper are in the set; otherwise, they are not. Given one object, how many states can the sheet have? How about with 2 objects? 3? 4? 5? N? How do you know? The set notation is for us. $\endgroup$ – Z4-tier Jan 17 at 23:53
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One thing that can't be shown without proof techniques is the impossibility of something. One reasonably concrete example of this is the impossibility of constructing a square and a circle with the same area using a straight edge and a compass. Individual cases can be shown to not work and to demonstrate that it's a hard problem, but there's a world of difference between hard and impossible.

Furthermore, there's a big difference between knowing that something works, and knowing why something works. A good proof not only demonstrates that a given property or equation works but provides an understanding of why it works.

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    $\begingroup$ "the impossibility of constructing a square and a circle with the same area using a straight edge and a compass" If the person doesn't understand division, and are shaky at anything beyond addition and subtraction, there is a good possibility they don't even know how to calculate area at all, let alone the area of a circle. $\endgroup$ – chasly - supports Monica Jan 16 at 12:05
  • $\begingroup$ You don't need to be able to calculate areas in order to have a feeling for what area is. $\endgroup$ – Vercassivelaunos Jan 17 at 13:42
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Theorem: Suppose you start with the number 1, and you're allowed to multiply it by 2, 3, and/or 5, as many times as you like. In this way, it's possible to get any whole number.

Proof: Check this for 1, 2, 3, 4, 5, and 6. Also true for other random examples I think of, such as 10, 100, and 96. It's clearly true.

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    $\begingroup$ There's a whole genre of jokes based on different types of people proving that ‘All odd numbers are prime’. (My favourite's the computation linguist who says “3 is an odd prime, 5 is an odd prime, 7 is an odd prime, 9 is a very odd prime, 11 is an odd prime…”) See e.g. here. $\endgroup$ – gidds Jan 15 at 17:45
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    $\begingroup$ I prefer the economist's version: "3, 5 and 7 are prime. Admittedly 9 isn't prime, but there must have been some special factors that interfered with collecting the data for that case, so we can ignore it". $\endgroup$ – alephzero Jan 16 at 2:25
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    $\begingroup$ This is a person who does not understand division and probably has trouble with a complicated multiplication. This would just seem like a nightmarish piece of homework for them. $\endgroup$ – chasly - supports Monica Jan 16 at 12:03
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    $\begingroup$ @Chasly it's still one of the answers that aligns more closely with OP's requirement. It could be fairly easily demonstrated using beads or pebbles... in fact for something like this I'd suggest that having to use a calculator would be counterproductive. $\endgroup$ – Mark Morgan Lloyd Jan 17 at 12:51
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Maybe you can use some non-mathematical examples...

Black swans. Europeans assumed that all swans were white, because all swans in Europe are white, to the extent that a black swan was considered impossible--like a "flying pig". When they finally got to Australia and discovered that there are black swans, it was a big shock.

Suppose someone tells you that they leave their car door unlocked every day and nothing bad has ever happened. How confident do you feel in leaving your car door unlocked? Do you think it's impossible for your car to be burgled, just because your friend's car hasn't been up until now?

Even better would be coming up with some examples relating to some area that your friend is interested in.

Then you can show some concrete things that can be proven... Drawing four aces in a row from a normal deck is unlikely, drawing five is impossible.

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Because mathematicians want to be really, really certain.

There are lots of ways you can be convinced that something is true, and be wrong. By requiring proofs before you are convinced, you can make that amazingly less likely.

A proof requires that someone lay bare all of the weaknesses in their claim, and show how those weaknesses are not a problem. Someone checking a proof can look at it, and even without understanding it fully, can determine if the person writing the proof cheated or made a mistake.

Imagine if you had a laser pointer, and when you pointed it at someone telling a lie, it was red, and if they told the truth, it was green.

And it wasn't only "the truth as they see it", but the absolute truth. If they think the moon is made of green cheese, and you point the laser at them, and they say "the moon is made of green cheese" it will show "that is a lie".

That superpower would be very useful. Proofs in mathematics give other mathematicians that super power. They can even point the laser at themselves, and see if their own statements are true or not!

Now mathematical proofs isn't quite as good as that laser, because the super power only works on proofs, not on everything you say, and checking a proof takes work (it isn't instant). Plus, finding proofs for something can be extremely hard.

Mathematics relies on this to do some of its amazing things. Mathematics builds these huge complex castles of thought, where each brick, each beam, each blob of cement is more mathematics. It is only because we can rely on the quality of the building-blocks that mathematics can build the complex stuff it can do.

To see how this is useful, imagine that laser pointer again. And you want to build the best castle. So you walk around and say "this rock would be good for building that castle", "I should build a wall here for my castle", "this wall is strong enough to hold up my castle", then check the laser pointer each time. That super laser pointer can teach you how to build the castle you want to build, even if you don't know how to.

Those castles of thought let you do things like predict the weather (with imperfect accuracy, but way better than we could even 10 years ago), build skyscrapers, split the atom, fly to the moon, or invent a covid-19 vaccine.

Now, not everything in those tasks is done with mathematics. But supporting all of that science are huge castles of mathematics that are relied upon to not be made of nonsense.

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An example that directly contradicts the idea of "seeing that an equation works after testing it for a few values:"

Theorem (not): The polynomial $x^2-x+41$ is prime for every integer value of $x$.

This works after testing it for every integer $x = 0, 1, 2, \dots, 40$.

So "obviously" (according to your friend!) its value is a prime number for any integer value of $x$. Just don't try $x = 41$.

Historical note: this was originally discovered by Euler, and mentioned in a letter to Bernoulli in 1771.

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    $\begingroup$ This is my favorite, and I use it 1st to introduce my discrete math course. However, I use a "smaller" version: $x^2 - x +11$. $\endgroup$ – Daniel R. Collins Jan 16 at 0:27
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    $\begingroup$ You have to be joking. Someone who can't do division will have no idea what a polynomial is. They won't have studied algebra even at the most basic level. You can bet your life they can't define a prime number, even if they have heard the word. $\endgroup$ – chasly - supports Monica Jan 16 at 11:58
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Put 100 red marbles in a box and 1 white marble. When you blindly pick a few marbles and they are all red, it does not mean all marbles in the box are red.

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  • $\begingroup$ No one would expect it to mean they were all red. How does this say anything about proof? $\endgroup$ – chasly - supports Monica Jan 16 at 12:23
  • $\begingroup$ @chasly-supportsMonica I think the idea is that by picking several marbles, one might think all the marbles are red. The only way to prove whether they are or not is to look at every single one. The concept of proof is pretty basic and doesn't require fancy epsilons and deltas and stuff. Anyway, +1 for me and I'm excited to see whether the user marbles plans to refer to marbles in every one of their future answers! $\endgroup$ – Thierry Jan 17 at 3:02
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    $\begingroup$ This is the most useful answer in my opinion, you don't need any math to understand it. A test is like blindly picking one marble. A proof is opening the box and looking at all of them. This should immediately show the value of a proof. $\endgroup$ – Guntram Blohm Jan 17 at 11:12
  • $\begingroup$ I'm rethinking this. My objection relates to what the subject believes beforehand. Some gamblers believe in a winning streak, "I've thrown 3 sixes, so the chances are, I'll throw another because my luck is in". Others go the other way, "I've thrown 3 sixes so my chance of throwing another is getting less and less". Both of these are fallacies. Probability is a tricky subject. You haven't sufficiently defined the parameters in my opinion. (i.e with or without replacement, prior knowledge or expectation about the colours, the number of marbles). The number is ... $\endgroup$ – chasly - supports Monica Jan 17 at 11:31
  • $\begingroup$ ... particularly important. If I offer you a bag containg 3 marbles and you draw one red, who knows what your expectation will be. If I offer a million marbles and you draw one hundred red, who knows what a non-mathematician's expectation will be? It depends on which superstition they subscribe to. $\endgroup$ – chasly - supports Monica Jan 17 at 11:34
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Most of the answers assume, as does the OP, that the purpose of proof is verification. In fact "establishing the truth of a proposition" is only one of the reasons that mathematicians prove things.

The most-cited paper on this subject is probably Michael de Villiers' "The Role and Function of Proof in Mathematics" (Pythagoras, 1990, pp. 17-24). De Villiers argues that, in addition to verification, proof serves all of the following functions:

  1. Proof is a means of explanation:

Although it is possible to achieve quite a high level of confidence in the validity of a conjecture by means of quasi-empirical verification (e.g. accurate constructions and measurement; numerical substitution, etc.), this generally provides no satisfactory explanation why it may be true. (emphasis in original)

  1. Proof is a means of systematization:

... proof which manages to expose the underlying logical relationships between statements... is an indispensable tool in the systematisation of various known results into a deductive system of axioms, definitions and theorems... Some of the most important functions of a deductive systematisation are given as follows...

  • it helps with the identification of inconsistencies, circular arguments and hidden or not explicitly stated assumptions
  • it unifies and simplifies mathematical theories...
  • it provides a useful global perspective or bird's eye view of a topic...
  • it is helpful for applications both within and outside mathematics, since it aids checking the applicability of a whole complex structure or theory simply by evaluating the suitability of its axioms and definitions
  • it often leads to alternative deductive systems which provide new perspectives and/or are more economical, elegant and powerful than existing ones
  1. Proof as a means of discovery

It is often popularly said by critics of the amount of deductive rigour at school level, that deduction in general (and proof in particular) is not a particularly useful heuristic device in the actual discovery of new mathematical results. Such people naively seem to believe that theorems are virtually always first discovered by means of intuition and/or quasi-empirical methods, before they are verified by the production of proofs... Perhaps this perception is due also in part to the stereotyped way in which proof is normally taught (first the result is presented followed by the proof). This view however is completely false, as there are numerous examples in the history of mathematics where new results were discovered/invented in a purely deductive manner... To the working mathematician proof is therefore not merely a means of a posteriori verification, but often also a means of exploration, analysis, discovery and invention.

  1. Proof is a means of communication

According to this view proof is a unique way of communicating mathematical results between professional mathematicians, between lecturers and students, between teachers and pupils, and among students and pupils themselves... Proof as a form of social interaction therefore also involves the subjective negotiation of not only the meanings of concepts concerned, but implicitly also of the criteria for an acceptable argument.

In case your friend is not persuaded by de Villiers' analysis, Yehuda Rav, in the aptly-titled article "Why do we Prove Theorems?" (Philosophia Mathematica, 1999, 5-41), considers the following thought experiment:

Let us fancifully pretend that the general decision problem did after all have a positive solution, that every axiomatisable theory was decidable, and that a universal decision algorithm was invented and implemented on oracular computers, marketed under the trade name of PYTHIAGORA (Pythia + Pythagoras). There she is, sitting on the desktop of every mathematician. Not only can PYTHIAGORA answer all our questions, but she does so at the speed of light, having no complexes about complexity of computation. Think how wonderful it all would be. You want to know whether the Riemann hypothesis is true or not, just type it in (using some appropriate computer language), and in a split second, the answer is flashed back on your screen: 'true' or 'false'... We mathematicians would only have to produce conjectures, and let PYTHIAGORA weed out the false from the true. What a paradise! What a boon!

Did I say boon? Nay, I say, no boon but doom! A universal decision method would have dealt a death blow to mathematics, for we would cease having ideas and candidates for conjectures. The main thesis to be developed here is that the essence of mathematics resides in inventing methods, tools, strategies and concepts for solving problems which happen to be on the current internal research agenda or suggested by some external application. But conceptual and methodological innovations are inextricably bound to the search for and the discovery of proofs, thereby establishing links between theories, systematising knowledge, and spurring further developments. Proofs, I maintain, are the heart of mathematics, the royal road to creating analytic tools and catalysing growth. (Emphasis added)

In other words: the activity of searching for and finding proofs gives rise to new ideas, new questions, and new connections between previously unrelated fields of research.

Seven years after Rav's paper, John Dawson wrote, in the same journal, a thematic sequel: "Why Do Mathematicians Re-Prove Theorems?" (Philosophia Mathematica, 2006, 269-286). As the title suggests, mathematicians do indeed devote time and effort to finding new proofs of already proved results, something that would make no sense if the sole purpose of a proof were verification! Indeed such proofs are sometimes highly regarded by the mathematical community; Dawson points out that "in 1950 a Fields Medal was awarded to Atle Selberg, in part for his elementary proof of the prime-number theorem."

Dawson identifies several reasons (and sub-reasons) for why mathematicians re-prove theorems. Here is an abbreviated version of his list:

  1. To remedy perceived gaps or deficiencies in earlier arguments. (Subreasons: to replace a non-constructive argument with a constructive demonstration; to provide a more efficient algorithm for performing a calculation; to eliminate controversial hypotheses, such as the parallel postulate, the Axiom of Choice, the Riemann Hypothesis, or the Continuum Hypothesis.)
  2. To employ reasoning that is simpler, or more perspicuous, than earlier proofs. (Examples: it may reduce the number of computations, or the number of cases to consider; it may be significantly shorter; it may require fewer hypotheses or fewer conceptual prerequisites.)
  3. To demonstrate the power of different methodologies (for research, pedagogical, or ideological purposes).
  4. To provide a rational reconstruction (or justification) of historical practices.
  5. To extend a result, or to generalize it to other contexts.
  6. To discover a new route.
  7. Concern for methodological purity.
  8. "Finally, the existence of multiple proofs of theorems serves an overarching purpose that is often overlooked, one that is analogous to the role of confirmation in the natural sciences...."

Obviously, given the limitations of your friend's mathematical background, not all of these explanations are going to be comprehensible or persuasive. But some of the main ideas should be clear regardless. In particular the fact that proof has an explanatory power, that it can be used to discover new things, and that it can be used to reveal the underlying assumptions we may be unconsciously making, should be persuasive reasons why proof has value beyond its power to verify.

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    $\begingroup$ This answer, while long and philosophic, keys in on what I think the critical element is: proofs are useful to some people because they do something. A person approaches the world differently because they have a proof. How they approach the world differently is a personal matter. An individual with very little math background unsurprisingly finds little value in the power-tool of mathematics. However, the people element can be related to them without invoking the mathematics. $\endgroup$ – Cort Ammon Jan 19 at 0:52
  • $\begingroup$ An excellent summary. Thanks! $\endgroup$ – vonbrand Feb 26 at 21:28
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From a practical point of view, your friend is right. One just has to be careful to test enough numbers. Engineers don't believe calculus because it has proofs based on analysis which is based on set theory; engineers believe calculus because lots of engineers before them used calculus to build bridges and very few of the bridges fell down. We have analysis in the first place, not primarily because of logical doubts about calculus, but because there were methods in summing Fourier series that were giving nonsensical results and people wanted to know why they were happening and what not to do to avoid them.

An alien civilization could treat mathematics entirely empirically, as a kind of empirical science of quantities (or even as a branch of physics), and, while this treatment has some disadvantages, they could have all the technology we have.

On Earth, Euclid decided that mathematics would be a good subject with which to teach and demonstrate the power of Aristotelian logic, and people ever since have found this approach to mathematics beautiful and attractive. It is one of the crowning glories of our civilization. Some of it may be hard for some people to understand, but no more so than Joyce's Ulysses or Schoenberg's Pierrot Lunaire.

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    $\begingroup$ Engineers have problems all the time...not necessarily the dramatic ones like a bridge collapse, but product recalls, new home defects, etc. All the time, even for products that were well tested and had no problems found, but if you have millions of people repeatedly using something they end up with unusual circumstances pretty often. Engineering doesn't skip proofs because testing is "good enough", it skips proofs because the systems are too complicated for humans to prove (yet). But I bet a lot of engineers would sleep a lot better if they could prove that their systems were good. $\endgroup$ – user3067860 Jan 15 at 20:11
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    $\begingroup$ @user3067860 - You can never prove your systems are good, because you can never prove that the mathematics involved is a faithful representation of the real world. You can model more and more of your systems, and prove theorems about your model, but the validity of the model is always in question. Mathematics studies mathematical models, not the world. $\endgroup$ – Alexander Woo Jan 15 at 21:39
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    $\begingroup$ "An alien civilization could treat mathematics entirely empirically..." Skeptical, citation needed. $\endgroup$ – Daniel R. Collins Jan 16 at 0:25
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    $\begingroup$ @user3067860 Even if you can't prove that a system is good, it's sometimes worthwhile to try. Once, I and some so-workers were asked to prove that a certain software specification ensured security. First, we had to make the specification precise (by asking the folks who wrote it what they meant in certain edge cases). Once that was done, it was clear that it didn't ensure security, and we could point out specific flaws. $\endgroup$ – Andreas Blass Jan 16 at 4:28
  • $\begingroup$ @DanielR.Collins "Luke, empirical mathematician, I am." $\endgroup$ – TripeHound Jan 16 at 7:41
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Try solving Pell's equation by trial and error. For example:


$x^2-61y^2=1$ has a whole-number solution $x=1$, $y=0$. Are there any others?


Try a few values for $x$ and $y$, and it will appear not. Is that a proof that there are no more solutions?

It turns out that the next smallest solution is $x=1766319049$, $y=226153980$, which would take quite a while to find by searching.

It's a relatively simple problem to explain - a square of tiles can be rearranged into $n$ smaller identical squares, with one left over - but for some values of the multiplier $n$ it leads to very big numbers. For other values (squares) there is no other solution - you would search forever.

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    $\begingroup$ You vastly overestimate a non-mathematician's knowledge. You suggest using algebra. Most people on the planet don't even know the word. I used to know someone who had had algebra lessons and still had no idea what it was about. They could do simple manipulations and get an answer but had no understanding. Literally, they complained to me, "I can do a bit of algebra but I never understood what x actually was. What is x?" $\endgroup$ – chasly - supports Monica Jan 16 at 11:53
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get him to draw a triangle (maybe even freehand), and measure the internal angles. point out all the things in the world built with triangles - electricity pylons, eiffel tower, maps etc - triangles are important, and its important in order to build our world to understand triangles completely.

what was the sum of the internal angles? (in my experience it won't be 180degrees due to inaccurate readings) Repeat it again with a few more triangles. Then explain that internal angles always add up to 180degrees - its been proved, so engineers dont need to guess or measure every triangle - they can just use the proved fact.

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    $\begingroup$ But the Indians and the Chinese built lots of things with triangles and also believed that their angles always added up to two right angles, long before the Euclidean notion of proof reached their civilizations. $\endgroup$ – Alexander Woo Jan 17 at 23:08
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    $\begingroup$ I like John's answer. I also think this comment is important. Perhaps they proved it to themselves somehow but never thought "proof" was important to record. Understanding multicultural approaches to mathematics is important. $\endgroup$ – Sue VanHattum Jan 18 at 2:39
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If you've got a sum you're keeping track of (e.g. you're adding up your purchases in a shop), you can quickly check whether your final answer is right by just repeatedly adding up the digits, ignoring any 9's, until you get a one-digit number, and then checking whether the sum of the digits of your sum is equal to the sum of the digits of the summands after doing the same thing to them.

For example, {£1.72, £5.99, £2.13}:

  • The sum is £9.84 on the one hand; ignoring nines, we get a digit sum of 8+4=12. The digit sum of 12 is 1+2 = 3, which is a single digit so we stop.
  • The digit sum of the components - ignoring nines - is {10, 5, 6}, which adds up to 21, whose digits add up to 3, which is a single digit so we stop.

So our test says "you didn't make a mistake as far as I'm concerned", because we got the same number 3 both times. If I'd got my sum wrong and I'd made £9.45, the digit sum of £9.45 (ignoring nines) would be 9, which (ignoring nines) has a sum of 0, and we'd know we had made a mistake either in the sum or in our digit-sums check.

This check is quicker and easier than just re-doing the sum, and as the list gets longer and the numbers get bigger, the technique gets cheaper and cheaper; and you can run the check incrementally, after every addition you perform.

In real life, this technique has a very long history in accounting folklore, so we can be quite sure it works; it's called "casting out nines". But what if we'd just come up with it ourselves? How could we know it actually works? It's extremely unobvious that the procedure gives you any meaningful answers. It's so unclear why it works that if I'd just discovered the technique for myself, I would be quite uncomfortable to use it. Only with a proof would I be happy.

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There is quite an amazing collection in response to the 11 yr-old MathOverflow question, Examples of eventual counterexamples. (This a precursor to the already cited 9 yr-old MSE question, Examples of patterns that eventually fail.) Here's one (among many) I like (posted by Gerry Myerson)

The numbers $12$, $121$, $1211$, $12111$, $121111$, etc., are all composite - until you get to the one with $138$ digits—that's a prime.

For example, \begin{align} 121 & = 11^2 \\ 1211 & = 7 \cdot 173 \\ 12111 & = 3 \cdot 11 \cdot 367 \\ 121111 & = 281 \cdot 431 \\ 1211111 & = 11 \cdot 23 \cdot 4787 \end{align} Here is the number with $137$ digits (parsed with commas): $$ 12,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111 $$ It is divisible by $3$ (and $11$, and $17$, and ...). Whereas the number with $138$ digits is prime: $$ 121,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111,111 $$

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    $\begingroup$ A person who does not know how to do division will not understand what a prime number is or how to find one. This is much too advanced. $\endgroup$ – chasly - supports Monica Jan 16 at 12:14
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    $\begingroup$ I used to use an analog of this (same idea, different digit sequence) as a 1st-day exploration in my community-college discrete math class for math and CS majors. Even there it was too complicated. The numbers quickly get too big to handle (even on a calculator), they don't have strategies for decomposing them or even verifying them, etc. Even sketched out a program for how to do a search in class and no one could ever successfully complete it. I've switched to the $f(x) = x^2 - x + 11$ function which involves much smaller numbers. $\endgroup$ – Daniel R. Collins Jan 16 at 15:00
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    $\begingroup$ @DanielR.Collins: Yes, probably too complicated. But still, to me: Amazing! $\endgroup$ – Joseph O'Rourke Jan 16 at 15:28
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Here's a surprising fact I've learned today on this very site:

Have your friend raise with a calculator

$1.5^{1.5^{...}} $.

At some point it goes to infinity.

Now let him do the same with

$1.4^{1.4^{...}} $

Funnily it stop growing. That raises some questions about the previous one : did it actually stop growing? Or was just the calculator screen to small and it stops eventually? And how can we ever be sure that's the case? And what about 1.49?

I guess you could do the same with the geometric series.

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    $\begingroup$ Mh. Raising is just repeated multiplication and it can be done with the calculator. $\endgroup$ – Three Diag Jan 16 at 12:46
  • $\begingroup$ If you don't want raising to fractional stuff you can do the geometric series as just a pattern, there is then plenty of real world examples to us such as accruing interests etc. $\endgroup$ – Three Diag Jan 16 at 12:48
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Let me be provocative.

Your friend is right. Nobody needs proof except mathematicians. The reason you want absolute proof is that you have a mathematical mindset. For all other people (farmers, carpenters, engineers, physicists) overwhelming evidence is sufficient.

It is telling that most non-trivial statements about actual nature cannot be proven to be correct, and don't need to be. For example all "knowledge" we have about our surroundings is fundamentally, utterly wrong. There is no solid matter, there is no absolute time and space. Newton and Leibniz were entirely mistaken (there is no Fernwirkung). A ball you throw does emphatically not follow a parabola, even in a vacuum; another categorical error. Of course it does not even follow an ellipse. In fact, nobody in the world can tell you exactly where it goes because the three body problem is unsolvable. That is quite typical for real-world problems: We can only provide numerical approximations. But our assumptions and approximations are often "good enough" (Malcolm Gladwell).

Proving mathematical statements is art pour l'art for people who like puzzles. Nobody really needs it. (This being said, if your proof provides an insight it has value. But the value is in the additional insight.)

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    $\begingroup$ Even worse, all these assertations are only about various models of reality. Even if a ball follows some definable curve, we could never prove it because (in so far as seems theoretically possible) we could never measure the path with zero error (quantum mechanics or no quantum mechanics). Indeed, for all we know, every particle in the universe "randomly repositions" every $10^{-24}$ seconds to some location (same or different) in the universe (nearby or light years distant), and we've just been living during a few thousands year period in which those repositionings (continued) $\endgroup$ – Dave L Renfro Jan 18 at 14:18
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    $\begingroup$ appear to follow the laws of physics that we know, and sometime tomorrow the apparent order vanishes and the "random repositionings" will return (not that they ever vanished, as we've just been living during a period when, so to speak, there was an incredibly long streak of "heads" during an endless sequence of coin tosses). $\endgroup$ – Dave L Renfro Jan 18 at 14:23
  • $\begingroup$ I deleted the comment because it wasn't important. $\endgroup$ – James S. Cook Mar 1 at 15:32
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To some extent, your friend is totally right. If you want to check if an equation always holds, then just checking several values is often a nice and practical way to do that.

But proofs can certainly come in handy sometimes. Here's a simplified version of a problem I encountered at work this week.

We had 8 cables of various lengths and 8 conduits of various lengths. We needed to put one cable in each conduit. Also, each cable needed to be longer than the conduit it was in. If a cable was too short, it wouldn't connect the two components on the ends of the conduit, so that would be bad. On the other hand, if a cable was too long, we could simply tuck the excess away, and that would be perfectly fine.

Here were the lengths of the conduits: 50 feet, 60 feet, 65 feet, 70 feet, 80 feet, 80 feet, 100 feet, 120 feet.

And here were the lengths of the cables: 55 feet, 60 feet, 60 feet, 70 feet, 75 feet, 85 feet, 110 feet, 130 feet.

If you simply start trying different ways of arranging the cables and conduits, you'll find that none of the ways you're trying work. But there are over 40,000 possible assignments of cables to conduits. Even if you try 1,000 arrangements and none of them work, how do you know that no solution exists?

Fortunately, we don't have to rely on trial and error. If you know the right "trick," then it's easy to prove that there really is no way of making these cables fit in these conduits. That's a lot easier.

(I sent an email to the project manager stating that the cables we had weren't long enough, and that we needed some longer cables delivered.)

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    $\begingroup$ But you don't need the notion of proof here. You just need someone to have passed down the rule that, if you put the lengths of the conduits in order and the lengths of the cables in order, the cable length must be longer than the corresponding conduit length. This rule could be observed empirically and taught. $\endgroup$ – Alexander Woo Jan 17 at 23:11
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It is possible to tile (tessellate) a plane with any triangle.
It is possible to tile a plane with any quadrilateral.
It is possible to tile a plane with some pentagons (such as 'houses' where the sides all have equal length, the base is a square, and the top is an equilateral triangle).
It is possible to tile a plane with some hexagons (such as regular hexagons).
It is impossible to tile a plane with any convex polygon with 7 or more sides. [1]

Depending on how open to being convinced your friend is, you could present a very contrived example like the question 'is n+100 > n+n for all integer n?'

As a side-note, you can argue against induction in general with a parable like that of the inductive turkeys. full version *I believe the parable originated from Betrand Russell, but I don't know the original source
The short version is that for the first few years of a farm animal's life, the farmer showing up means food. But then one day without warning it means the farmer has come to kill them.

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    $\begingroup$ I'd suggest that somebody who has problems with fundamental arithmetic would balk at the idea that tiling a surface is in any way related to mathematics. In fact, what defines whether something is mathematics other than that mathematicians are interested in it? $\endgroup$ – Mark Morgan Lloyd Jan 17 at 12:46
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since it's enough to see that an equation works after testing it for a few values.

  1. Would they be willing to bet their life on that if that equation was being used for an autopilot in an airplane or some other piece of equipment? Or if the equation takes millions of possible combinations of values?

  2. Suppose I make an engine. I test it and it works. Great. Now what? I can make that same engine again, but without understanding how it really works I will have a lot of trouble improving it and building upon it. Proofs are the mathematical equivalent of picking something apart to see if and why it is true.

  3. Related to number #2, math is not really about the investigation equations. It is about the investigation of concepts and ideas of which equations are a very tiny part. Imagine a biologist comes up with a theory: "All birds fly." So he finds a few birds and they all fly. Does that mean he is correct? Were his few tests good enough? We know that they were obviously not because a few odd cases were missed.

    Those few odd cases that were missed can cause big problems if another mathematician assumed it was true and used it in their own work for something else, or can cause literal disaster if it is being used to calculate the mechanical stresses of a bridge or used in a computer program that controls heavy machinery.

    For a biologist to really prove that he would need to find every single bird in existence. That's obviously impractical, and even when talking about simple equations, there are far far more numbers, let alone combinations of numbers that can go into an equation. But unlike the biologist, in math you have the luxury that it follows logic so you can actually verify whether something always works or whether it only works under certain conditions and figure out what those conditions are. Those are proofs.

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  • $\begingroup$ If they can't do division, they won't know what an equation is. Algebra is far more advanced than arithmetic and relies on a facility with arithmetic including division. $\endgroup$ – chasly - supports Monica Jan 16 at 12:20
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    $\begingroup$ @chasly-supportsMonica: The claim about equations from the OP asserts that it comes from the person in question. $\endgroup$ – Daniel R. Collins Jan 16 at 15:04
  • $\begingroup$ @chasly-supportsMonica The person asserting the position is making the claim about equations. If you are trying to convince someone with a lack of knowledge you would be best off to try and do it on their terms, rather than your own, particularly if your own means introducing a bunch of stuff they don't understand. $\endgroup$ – DKNguyen Jan 16 at 18:08
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Keep this person as a friend and admit -- if asked -- that you have no good explanation for someone unfamiliar with mathematics. (In general, spouses of mathematicians never ask!) If your friend plays bridge or chess, discuss the strategies of the game. You could then say that proofs give mathematicians the guarantee that their strategies will work. Engineers have methods that (almost) guarantee that their structures will not collapse.

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Now, he asked me why you'd need proofs for anything ever, since it's enough to see that an equation works after testing it for a few values.

So let me give my take on a practical example which does not involve numbers or formulas, so it should be relative simple to understand.

Having a mathematical proof for some calculation is like having a map.

Let's say that you live in Village A and you want to visit Village B. There is a large forest in between and there has been no contact between the villages before so all you have to begin with is an approximate direction. Well, you will then have to walk out in the forest and try and try to find Village B. Assuming you find it and then put up marks along the way, that's like a formula discovered that other people can use. And if all you ever want is to travel back and forth between Village A and Village B, then by all means you do not need to do anything more.

But is it the fastest way? Is there an alternative route that does not have that many ups and downs over hills? What if part of the path becomes inaccessible due to landslides or floods?

If you have a map then you will be able to reason about alternatives outside the already existing path without actually having to resort to trial and error. Without a map you will just have to test out alternatives and see if they fit. So maybe you found a new path that is a bit shorter. Great, but is it the shortest? When can you stop testing to conclude you have found the shortest path?

Then maybe you want to visit Village C. If you have a map you are able to find out already before you start that it is impossible to walk to Village C because it is located far out on an island, whereas if you just set out walking trying to find it you might be caught up in an never ending quest doomed to fail.

Now the above village scenario is perhaps a bit contrived, but it is not extremely far away from the travelling salesman problem which is a real mathematical problem which has had a lot of research applied to it.

For this problem, theory/knowledge/proofs outside an actual formula/algorithm/implementation to find the shortest path tells us that there exists some upper limit where it is impractically impossible to find the optimal solution.

Thus the approach of just testing a formula with a few values to see that it works is not enough.

Q.E.D.

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Does your friend get bothered by multiplication? You say $+$ and $-$ are okay but division is problematic, so you skipped over multiplication.

If your friend is okay with multiplication, and knows about breaking up numbers into prime factors, here's something to claim: every number has at most three different prime factors ($45 = 3 \cdot 3 \cdot 5$ has just two different prime factors, for instance). This is true for all numbers up to $100$, but of course it is ultimately wrong and the first number with four different prime factors is $2 \cdot 3 \cdot 5 \cdot 7 = 210$. So a pattern that holds for $100$, or even $200$, examples need not hold all the time.

Or how about this: if $p$ is prime then $2^p-1$ is prime:

(1) $2^2 - 1 = 3$ is prime

(2) $2^3 - 1 = 7$ is prime

(3) $2^5 - 1 = 31$ is prime

(4) $2^7 - 1 = 127$ is prime

(5) $2^{11} - 1 = 2047$ is prime... just kidding: it is $23 \cdot 89$.

So the pattern broke after five tries. Now ask your friend: is some definite number of examples always good enough?

Claim: A number $n > 1$ is prime exactly when $2^{n-1}-1$ is divisible by $n$. This is correct for $n \leq 300$. It breaks for the first time at $n = 341$: $2^{340}-1$ is divisible by $341$ but $341$ is composite: it is $11 \cdot 31$. The next counterexample is $561$, which is $3 \cdot 11 \cdot 17$. There are infinitely many further counterexamples.

So again we have a pattern that works initially but eventually doesn't. The lesson here is that in math, merely believing something because it works for a bunch of examples is not a viable method of knowing what is correct and what is not correct.

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    $\begingroup$ "breaking up numbers into prime factors" This requires understanding of division. We have explicitly been told they don't understand division. You think they will understand raising a number to a power? They won't. $\endgroup$ – chasly - supports Monica Jan 16 at 12:19
  • $\begingroup$ @chasly-supportsMonica: Technically just multiplication, which as stated is a gap in the OP. $\endgroup$ – Daniel R. Collins Jan 16 at 15:08
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Money, even imaginary money is a good motivator

Much mathematical thinking historically, came from gambling and card playing. If you want to avoid losing your shirt, you need to understand the mathematics. It is a very practical use of the subject.

A simple experiment is to throw two dice. Ask your friend to bet on which sum will come up most often (use a box of matches each). Make sure you choose first and choose seven. Repeatedly throw the dice. You will win in the long term and your friend will become bankrupt. Ask if the dice are loaded. See if the other person can come up with an explanation of why certain sums are more likely than others. This will get them thinking.

They have observed that seven is the most likely sum so now ask them to prove that seven is the most likely result. This will help them to start constructing proofs.

Eventually you may nudge them towards drawing a diagram like the following.

enter image description here

Forget precise probabilites, just point out that seven is more likely because there are more ways to achieve it with two dice.

There are some good stories to be found online regarding gamblers being cheated by con-artists.

From here you can point out that people with no knowledge of maths are vulnerable to being manipulated with respect to money. Collect a few examples.**


** E.g. Pyramid schemes etc.

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    $\begingroup$ It doesn't sound like this will solve the stated question. Person can say, "Yes, from the experiment now we both know 7 is most common, no need for a proof". $\endgroup$ – Daniel R. Collins Jan 16 at 15:07

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