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Context: I know nothing about proofs and only a small amount about formal logic used in proofs. I'm trying to learn the basics of how to write a proof.

For example, suppose I wanted to prove that "all integers divisible by 4 are also divisible by 2".

My first reaction to this would be: "It's self-evident, because 4 is just 2*2". But this is obviously not a formal proof.

Another attempt: "Any integer n that is divisible by 4 must have 2*2 as part of its prime factorisation, therefore it is divisible by 2." But this seems to be explaining something simple by means of something more complex.

It seems that you are just supposed to know that "all integers divisible by 4 are also divisible by 2" and take it for granted. Or you are supposed to know, more generally, that all integers divisible by N are also divisible by P if N is divisible by P. Do you have to know this beforehand? Do mathematicians sometimes just say "it is true because it's obvious"?.

matt_rule
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  • Formal logic isn't the only way to write a proof. In fact, to me this is the least attractive. – Kurt G. Jun 11 '24 at 14:43
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    "But this seems to be explaining something simple by means of something more complex.": pretty much a summary of what mathematics looks like most of the times – Zima Jun 11 '24 at 14:47
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    Frankly, mathematicians do sometimes say "... because it's obvious", but this is only acceptable when you and your audience are at a sufficient level of mathematical maturity to synthesize the relevant proof in your head on the fly. Every such claim should be provable, but we don't have enough time in this life to fully write out every proof. So, this is an unfortunate but sometimes-necessary shortcut. When you are just starting to learn formal mathematics, you should avoid this as much as possible and prove everything you can! – diracdeltafunk Jun 11 '24 at 14:48
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    little steps don't get written out in proofs because the proofs are trivial and everyone reading the proof is supposed to be able to (easily) proof that themselves, just because it makes a proof very unreadable when writing every trivial step. But technically, every proof should be able to be written in terms of axioms. They don't say "it is true because it's obvious", because if you ask them, they can easily proof it and they know why it's true, they might say "it is obvious that it's true" but that is something different. – Vincent Batens Jun 11 '24 at 14:49
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    Btw, the argument you use makes use of the prime factorization, but doesn't have to: $4|n$ (read as $4$ divides $n$) means there exists an $m$ such that $4\cdot m=n$ but then also $2\cdot(2m)=n$ so per definition $2|n$. – Vincent Batens Jun 11 '24 at 14:50
  • Interesting question, which could be part of a mathematical culture that is not precisely abundant. – Ataulfo Jun 11 '24 at 14:55
  • I'd say the Reflexive Property of Equality (a number is equal to itself or a shape or line segment is congruent to itself). –  Jun 12 '24 at 00:41
  • It's basic but still requires proof, such that it's a common textbook-presented theorem. For example (replacing 2 and 4 with more arbitrary numbers), in Rosen's Discrete Mathematics and Its Applications, this is Ch. 4, Theorem 1(iii), with the proof left to the reader as section Exercise 4. At the level you're working with, you should always start with, e.g., "What is the exact definition of being divisible?" – Daniel R. Collins Jun 12 '24 at 06:31
  • you might enjoy reading Lectures on the Philosophy of Mathematics by Joel David Hamkins, there is chapter "5. Proof" about all this – jhole Jun 12 '24 at 12:20
  • I think this is a good example: Henri Poincaré observed that there seems to be no way to justify the principle of induction, other than by much more complicated arguments that (perhaps implicitly) depend on induction anyway. Here are Thoralf Skolem's thoughts on this. – MJD Jun 12 '24 at 15:38
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    Here's important question to consider. This statement does indeed have a formal proof. Do *you* know how to write one? If so then, when giving a mathematical exposition on the topic, you can use your own judgement regarding whether the proof is sufficiently simple that you can omit it (or leave it as an exercise). But if not, then you had better learn how to do it!! – Lee Mosher Jun 12 '24 at 16:13
  • I always felt this way about the inclusion-exclusion principle. To comprehend the formula you have to already know the proof (I think). – preferred_anon Jun 13 '24 at 08:24
  • The proofs you've offered suggest you could benefit from thinking more carefully about divisibility. Try to prove this generalization: "if $a$ divides $b$ and $b$ divides $c$, then $a$ divides $c$." This is "obvious" in that you've known it your whole life, but a careful proof of it is a good first exercise in unwinding definitions. – Charles Hudgins Jun 13 '24 at 15:29
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    Long ago many thought that unique factorization of integers was so obvious that no proof was needed. This was not corrected until 1801 when Gauss plugged this gaping logical gap in his famous book Disquisitiones Arithmeticae, where he wrote "It is clear from elementary considerations that any composite number can be resolved into prime factors, but it is often wrongly taken for granted that this cannot be done in several different ways". See also unique steps leading to a nonunique answer. – Bill Dubuque Jun 14 '24 at 07:33
  • I get goosebumps when I see someone saying "it's obvious" in math. I've probably seen too many counter-intuitive math riddles. – Jeyekomon Jun 14 '24 at 12:08

5 Answers5

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Great question, and well done for thinking carefully about the basics of proof and mathematics. Let's get into it!

It seems that you are just supposed to know that "all integers divisible by 4 are also divisible by 2" and take it for granted.

This is not true! We can and must prove it.

In order to prove a statement, we must know what it means. In this case, we must define "divisible".

Definition. Let $a$ and $b$ be integers. We say that "$a$ is divisible by $b$" if there exists an integer $c$ such that $a = bc$.

Now that we know what the statement means, we can prove it!

Proposition. Let $a$ be an integer. If $a$ is divisible by $4$, then $a$ is divisible by $2$.

Proof. Suppose $a$ is divisible by $4$. This means that there is an integer $c$ such that $a = 4c$. Equivalently, $a = 2(2c)$. Since $c$ is an integer, $2c$ is an integer. Thus, $a$ is divisible by $2$. $\square$

Of course it would be annoying to reprove thousands of different statements like this for every instance of a basic divisibility fact we need to use. So, it's much better to prove general statements like the one you suggested. Can you try to prove the following more general statement?

Proposition. Let $a$, $b$, and $c$ be integers. If $a$ is divisible by $b$ and $b$ is divisible by $c$, then $a$ is divisible by $c$.

Proof. Fill me in!

Everything in mathematics is this way. When we want to know that something is true, we must first make careful definitions to specify exactly what it is that we want to prove. Once we have those definitions in place, we can try to prove whatever we want!

diracdeltafunk
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    Thanks for the answer! As for the problem, I stared at this blankly for a long time, and had a lot of failed lines of thought. At the moment it's hard for me to think in this way, at least quickly. But here's my attempt, using the "divisible" definition you provided above: Proof. Suppose a, b and c are integers where a is divisible by b and b is divisible by c. This means that for some integer d, b=cd, and for some integer e, a=be, and further, that a=cde. Because c is a factor in this product, a is divisible by c. – matt_rule Jun 11 '24 at 15:45
  • I tried a proof by negation and realised that it was the same difficulty as proving things the positive way. And then I had this argument: '...This means that for some integer d, b=cd therefore to state "a is divisible by b" is to state "a is divisible by cd" and therefore "a is divisible by c" ', but I felt I'd only get half marks for that. – matt_rule Jun 11 '24 at 15:50
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    @matt_rule Reword "is a factor" sentence to something like "$a = (cd)·e$ implies $a = c·(de)$, which means $a$ is divisible by $c$", and you got it. – Paŭlo Ebermann Jun 11 '24 at 23:36
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    "We can and must prove it" - that very much depends on the context. Not every discussion of mathematics, even in classrooms, is part of a careful systematic development. In this case, yes, the context "want to prove that" deserves something more than "it's obvious" and more concrete than intuitive. – aschepler Jun 12 '24 at 01:43
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    In a fully rigorous proof of this, you would explain that the "equivalently" step is valid because multiplication is associative. – Barmar Jun 13 '24 at 22:49
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No

Nothing is "self-evident" in mathematics, although this is a relatively new position (circa. 150 years). For millennia, mathematicians considered axioms (or postulates) to be self-evident, but now we consider them to be unprovable assumptions that are accepted so we can do mathematics, and each branch of mathematics may (probably will) be based on different axioms.

For example, the postulates of Euclidian geometry are:

  1. Given any two distinct points, there is a line that contains them.
  2. Any line segment can be extended to an infinite line.
  3. Given a point and a radius, there is a circle with center in that point and that radius.
  4. All right angles are equal to one another.
  5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (The parallel postulate).

Now, for centuries, the parallel postulate bothered people - it's not pithy like the other four, and it doesn't seem quite as "self-evident" - surely, it can be proved from the other four? But it can't be. If you take it away, you get a different branch of mathematics, non-Euclidian geometry, where, among other things, the interior angles of triangles don't add up to 180o. To a large extent, it was the development of non-Euclidian geometries that led mathematicians to the realisation that axioms aren't "self-evident" in the same way that the rules of Association Football aren't self-evident; they're just the rules you have to accept if you want to play the game.

A mathematical statement can only be of the following kinds:

  • An axiom, postulate or assumption which, as stated, is accepted as true to form the basis of the discipline.
  • A theorem is a statement that has been proved from the axioms. Lemma is a minor theorem you prove on your way to a bigger theorem. A corollory is a theorem that pops out as a by-product; for example, in proving two lemmas, a third may also be proven by combining the first two with no further steps.
  • A conjecture is a proposition that is offered on a tentative basis without proof. A conjecture may be resolved by proof (becoming a theorem), disproof (becoming garbage), by being accepted as an independent conjecture (becoming an axiom in a new branch of mathematics), or it may be undecidable (from Godel's incompleteness theorem - unfortunately, for most conjectures, it's undecidable if they're undecidable). It is possible (and done) to build whole branches of mathematics based on a conjectures in the hope that someday someone will prove it.

Since you're working with natural numbers, you need to know the Peano axioms for the natural numbers:

  1. 0 is a natural number.
  2. For every natural number x, x = x. That is, equality is reflexive.
  3. For all natural numbers x and y, if x = y, then y = x. That is, equality is symmetric.
  4. For all natural numbers x, y and z, if x = y and y = z, then x = z. That is, equality is transitive.
  5. For all a and b, if b is a natural number and a = b, then a is also a natural number. That is, the natural numbers are closed under equality.
  6. For every natural number n, S(n) is a natural number, where S is the single-valued successor function (i.e. S(n)=n+1). That is, the natural numbers are closed under S.
  7. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.
  8. For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.
  9. If K is a set such that: (a) 0 is in K, and (b) for every natural number n, n being in K implies that S(n) is in K, then K contains every natural number. That is, any natural number can be obtained applying the successor function sufficiently many times to 0.

None of these is "self-evidently" true; they are just the rules we need to define what the natural numbers are. Further, they are insufficient to define the integers, the rationals, the reals, or the complex numbers - each of those needs additional axioms. They're also of little assistance in defining geometry - Euclidian or otherwise.

From these axioms, we can define addition and prove its commutativity. Given addition, we can define multiplication and prove that it is commutative and distributes over addition. Once we have that, we can prove your proposition.

However, it isn't necessary to prove everything from the ground up every time - you are allowed to rely on theorems proved by others. For example, @diracdeltafunk's proof assumes multiplication, which is a perfectly reasonable assumption, but it also presumes that you understand the definition of multiplication over the natural numbers, again, a perfectly reasonable presumption. However, to be a really rigorous proof, it should state that it relies on the definition of multiplication over the natural numbers.

Dale M
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  • Thanks for the background and for exploring the fundamentals of the question! You've given me a lot of things to research. – matt_rule Jun 12 '24 at 09:55
  • Is the list of mathematical statement types complete?  (For example, ‘1 = 0’ is clearly neither an axiom/postulate/assumption, nor a theorem/lemma/corollary, nor a conjecture.  And would ‘let x = 1’ count as an axiom, since it's clearly neither of the other types?) – gidds Jun 12 '24 at 13:21
  • I disagree with your claim that "nothing is self-evident in mathematics". Henri Poincaré and others opined that your (9) was self-evident, and that it must be taken as an axiom because there is no way to prove it that is not circular. One can construct a model of the natural numbers, say in ZF, and then prove that (9) holds for the model. But the axioms of ZF are a great deal more complex than your 1—9, and any proof that (9) holds in the ZF model will depend on (9), many times over. For more details see this post quoting Thoralf Skolem. – MJD Jun 12 '24 at 15:42
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    @MJD A counterexample: Let K be the set of natural numbers as we know them plus the numbers A, B, C, D where S(A)=B, S(B)=C, S(C)=D, and S(D)=A. This meets all of axioms 1-8 but violates 9. Without 9, the other axioms allow closed loops of numbers outside the normal counting numbers. – Dale M Jun 13 '24 at 02:53
  • @gidds no, its not complete - its only dealing with for want of a better word "conclusion" statements. Also '1=0' is a disprovable postulate. – Dale M Jun 13 '24 at 02:54
  • Sure, but the question here is whether the axiom is true of the actual natural numbers. Of course one can make up all sorts of objects with no interesting properties at all. – MJD Jun 13 '24 at 03:05
  • @MJD The axioms define the natural numbers - that's the point of axioms. Without 9, they also define things that are not the natural numbers. – Dale M Jun 13 '24 at 03:09
  • No, the axioms define a model for the natural numbers. There were natural numbers long before there were Peano axioms. That's why we bother to prove things like $\forall a.\forall b. (a+b) = (b+a)$. We already know that addition is commutative. We prove the theorem to show that the model accurately reflects what we already know about natural numbers. – MJD Jun 13 '24 at 12:45
  • You left out the thing most important to the question, what a proof is. – philipxy Jun 13 '24 at 19:22
  • @MJD the natural numbers, like all mathematics are a human invention - there aren’t 5 sheep on the hill unless there’s a human there to count them. – Dale M Jun 13 '24 at 22:06
  • @MJD, Dale, I think you are both correct. Mathematicians invented the natural numbers. But they did not come up with it out of the blue. They invented it to model something which was already in use, the counting numbers. While counting is again a human invention I don't think that counts as a mathematical invention as opposed to a 'cultural invention'. – SweepingBishops Jun 14 '24 at 08:04
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There's an old joke, sometimes claimed to be about Wolfgang Pauli, of a mathematician who says "It is obvious that ..." and when questioned "Is it really?" spends a large amount of time thinking about it (possibly even scribbling down an extensive proof) before finally saying "Yes, it is obvious."

The things we do in mathematics need to be rigorous, and when mathematicians looked at some of the things that were previously taken for granted they found that actually not only were they not obvious, they were potentially broken at quite a fundamental level.

There is another joke that it took 1000 pages to prove that $1+1=2$. The Principia Mathematica is a book that attempts to build mathematics from the ground up, with complete rigor, starting with the very concept of what it even means for a statement to be provable, and the language required to even say what $1+1=2$ even means. Once that's established, actually proving that it's true - in this particular framework - really only takes a few lines.

Once something has been proven, though, it is essentially "free fodder" for future mathematics. If I need to use $1+1=2$ in my own proof, I don't have to write out the entire Principia, I can just say "$1+1=2$ (Russell and Whitehead, 1913)". In fact, I usually only need to include the reference if it's a statement that isn't already considered fundamental to the subject I'm working in. So $1+1=2$ is at this point considered a baseline fact in pretty much any branch of mathematics that might need to refer to it, just like the statement "If $p$ divides $a$ and $a$ divides $b$ then $p$ divides $b$". But that still relies on having a clear understanding of what that all means and the context where we're using it. There are even times when a proof will need to use a smaller result (called a lemma) and the author may only sketch out a basic proof of it if the audience is expected to be able to fill in the blanks themselves.

If you take undergraduate-level courses in topics like logic, proof writing or number theory then you will wind up proving a lot of these apparently obvious things, and it will give you an appreciation of how important it is to be able to do this because mathematicians like to generalise common structures (things like groups, rings and fields are generalisations of sets like the natural numbers and real numbers) and not every proof carries over unscathed. You can have a structure where "$p$ divides $a$" is a meaningful statement, but is it still true that "$p$ divides $a$ and $a$ divides $b$ means $p$ divides $b$"? Better pull up the proof of that in the natural numbers and check whether it still applies here.

So to answer your question, yes it is obvious.

ConMan
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    To get a little deeper in the philosophy, $1+1=2$ was of course well-known and "obvious" long before Russell and Whitehead. So maybe more signficant than that proof justifying $1+1=2$, the proof shows that their logical system does model some expected and familiar math. Likewise whether working in Peano, ZF[C], or something just invented, if a theory gives meaning to $=$, $1$, $2$, and $+$, then it ought to model $1+1=2$ (or note how/when/why things differ). – aschepler Jun 12 '24 at 01:57
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    @aschepler: So it's the mathematics equivalent of implementing an entire microkernel from scratch, and then writing a hello world program to check that the microkernel works (for a certain bare minimum value of "works," anyway). – Kevin Jun 12 '24 at 02:15
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    @aschepler: Yeah, the nature of formalization has evolved. These days, I might show somebody a proof that 2 + 2 = 4 instead. It's still interesting that 1 + 1 = 2, but it's tautological because we usually define 2 as 1 + 1; see the definitions of 2, 3, and 4 in the linked proof. – Corbin Jun 12 '24 at 05:05
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The higher the level, the more synthetic the proof of a proposition or theorem is, which is due to practical and situational reasons. At elementary level one needs to be explicit enough while at high level in general the proof is short, incomprehensible to beginners because it is based on background knowledge.

This does not prevent a long proof from being significantly shortened using another route. An example of this is given by Mordell's and Weil's proofs of the Mordell's discovery of the finite basis theorem for elliptic curves (Weil's proof is shorter and simpler than Mordell's but both proofs are rigorous). Another very notable example is given by the proofs of irresolvability with radicals of equations of degree greater than or equal to $5$, given first by Abel and then by Galois, the first being very intricate and the second elegantly concise.

Ataulfo
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Tangential to this discussion, Poincare's book: science and hypothesis has a great presentation in chapter 1 on "the nature of mathematical reasoning". It addresses the question considered here: "what is mathematics, what is a mathematical proof, what is the necessary rigor that underlies it, etc" and should be readable by anyone who understands recursion and arithmetic.

Guillaume Dehaene
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