How do we know if a mathematical proof is valid?

Question

Georg Cantor has showed there are more real numbers than natural numbers in his diagonal argument. Assuming that two sets have the same size if we can make a pair up elements from set A with elements from set B. Now if we make a list of natural numbers, then no matter what list of real numbers we provide, there will always be another number that's not on the list (created by adding '1' to diagonal of the list, for example).

Now, it's a proof that there are more reals. But what if I told you, before Cantor, that there are equally many reals and natural numbers, because we can make a list of natural numbers and assign a real number to each natural number? I will tell you, look, it can't be the case there are more real numbers, because there are infinitely many nautral numbers and you will never run out of natural numbers to pair them up with real numbers!

Would it be considered a valid proof? Actually, wasn't it considered a valid proof until Cantor came up with his theory? What if there are other such theorems in mathematics commonly believed to be true, with accepted proofs, and some day someone will come up with a counterexample, disproving them? Does it mean we can never be sure if a mathematical proof is valid?

Answer 1

Some comments.

1) Well before Cantor, it was already known that we may "have troubles" in comparing infinite collections of numbers ; see, at least, Galileo's paradox.

2) Of course, the purported proof : "that there are equally many reals and natural numbers, because we can make a list of natural numbers and assign a real number to each natural number" is simply wrong. We may pair, e.g. the natural number 2 with the real number 2.0, of course, but we will have in any case "unpaired" numbers like sqrt(2).

3) The history of mathematics is full of "wrong" proofs : they have been corrected. See for example Girolamo Saccheri's proof of Euclid's fifth postulate in his Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw - 1733).

4) Also if it is a "minority" part in the mathematics community, there are some mathematicians who do not agree on the validity of some methods of proof commonly used by the "mainstream" mathematicians; see Intuitionism in the Philosophy of Mathematics and Intuitionistic Logic.

Answer 2

Your argument is not valid for the following reason :

It is not enough to show that you can assign a real number to each natural number. You must also show that you can assign a natural number to each real number.

Logically, you must show A implies B and B implies A in order to prove that A and B are equivalent.

Infinity produces several truths contrary to our immediate intuitions. The apparent fact that one infinity may not be "big enough" to accommodate another infinity is just one example.

Answer 3

There was Alfred Kempe's (in)famous proof of the Four Colour Theorem given in 1879, which was widely accepted and only shown to be incorrect in 1890. (Almost the exact same proof is used to show the weaker result that every planar map can be coloured using only five different colours, and that version is correct). (And importantly for mathematicians but not for this discussion, the flaw in the proof was fatal and couldn't be fixed; later correct proofs were much more difficult and completely different).

So taking that proof as an example, for 11 years people "knew" that the proof was correct, but it turned out to be wrong. (Reading the proof, it was a rare situation where the flaw was reasonably obvious for a 21st century software developer who has a different way of looking at problems than a 19th century mathematician). It seems that in practice we cannot know that a mathematical proof is valid. In theory we could - if they had been clever enough, people in 1880 would have looked at Kempe's proof and spotted the flaw. But in practice, they were not.

Or look at Wiles' proof of Fermat's "Last Theorem". To me, it's a nice read. The words are put together nicely. It looks like really clever mathematics to me. I don't understand one word of it. (This is different from things written by the many crackpots with similar proofs; there are many where I don't understand one word either but they stink). I'm sure when he published it, he knew the proof couldn't be valid, because it is so complicated, it's impossible to get it right the first time. Mistakes were found, they were fixed, and now if there are any flaws making the proof invalid, there is nobody clever enough to find them. There may be no flaws. There is actually a good chance that there are no flaws. I don't think we can say we know it is valid. We can say that those mathematicians who understand the subject believe that there is a good chance that it is valid, and if not, there is a very good chance of turning it into a valid proof with few changes.

Answer 4

The question may gain distinguishing what kind of validity you are seeking for.

On the one hand, you may be "simply" interested in using the state of the art mathematical tools to ensure that your engineering project as a consistent specification and that the thing that you built map exactly that. If that's what you are interested for, proof assistant may be an interesting entry point.

On the other hand you may be interested with more ontological concerns, like knowing if mathematical objects reveals eternal underlying truths or if their are just mere mental representations resulting from an ugly patchwork of cognitive biases. If that's what you are looking for, you may look up for realism, idealism, solipsism as entry points.

Answer 5

Every mathematical proof is done within the scope of formal system. Such a formal system is:

• A set of symbols that can be used
• A grammar to define what sequences of symbols (sentences) are well formed
• A set of axioms
• A set of inference rules to go from one sentence to another

Over time, the formal systems that are accepted shift to accommodate new discoveries or problems with old results. In the case of the infinite sets you describe, it was the shift to the set of axioms towards modern set theories that lead Cantor to challenge the conventional thinking.

As a result, you could still write any of the old proofs, and claim them to be proofs, but they would no longer be of value because the newer set of axioms had become the "preferred" set of axioms. Proofs using other axioms were simply less useful.

We also see this today with the difference between proofs in ZF and ZFC. Mathematicians are divided about the validity of the Axiom of Choice (the C in ZFC). A proof done in the formal system of ZF's set theory is considered more valuable than a comparable proof done in a formal system of ZFC's set theory.

Answer 6

Surely people have made incorrect arguments that have been generally accepted as valid until they were examined more closely.

But I don't think your imaginary pre-Cantor example could have been one of them, because, when presented with that argument, the first thing any mathematician would have asked is: What exactly do you mean by "same size"? What exactly do you mean by "bigger"?

Then one of four things would have happened:

1) You might have given Cantor's definitions, in which case it would have been immediately obvious that your argument proves nothing.

2) You might have given some other definition which would have made it immediately apparent that your argument proves nothing.

3) You might have given some other definition according to which your argument is entirely correct (but not in conflict with Cantor's, because you and he are using the same words to mean different things).

4) You might not have had a precise answer, in which case the mathematician would have immediately known that you were talking nonsense.

Answer 7

I do not think I will post a definite opinion about the question debated. Instead I will express doubt of the very logical validity of a statement that there are so and so many this kind of numbers or that kind of numbers, at least not before one defines what one means by the words "is" or "are" related to numbers.

A number is not anything that exists before you have a situation where you can formulate a number that describes that situation quantitively. So instead of proving how many numbers you have, or there is, I think the only logically valid way is to prove in which situations you can formulate a certain kind of number.

Answer 8

In Mathematics we can find comparatively more concrete ideas. But some are rather abstract. In the case of the former one, if we got the proof by more than one method we can make sure that the proof is valid. Eg: We can prove the Pythagoras Theorem by different methods and many men have already 'lived with it' and solved the problems in their real life situations. So we know that Pythagoras Theorem is valid in our world. But this is not the case of some abstract ideas like real numbers, complex numbers etc.

When mathematics is concerned we have to deal all its different branches as a whole.

In the case of Mathematics, to know, to make sure etc. means 'to get the proof of something'. I don't wish to keep any suspense in your reading my answer.

In this case, a mathematician wouldn't be a much better analyzer than a layman. Both of them would say: "It is impossible".

We never say that something is a proof if it can't be explained clearly at least to one higher authority. Even though someone says he got a proof that couldn't be explained, we wouldn't admit it. We would hesitate even to say that he knows it.

Mathematics is stubborn in its approach. So I will have to deal this question mathematically / rationally.

Mathematicians will often expand this question up to any extent like this (here, 'know if' means 'prove'):

Know if [Know if {Know if (Know if a mathematical proof is valid)}]

The number of persons who can 'get out of' even from the double bracket would be a finger-limit and eventually it reaches one. Then we will reach a situation as I mentioned earlier. So we can conclude that it is impossible.

Even if all the humans are transferred to another mathematical world, this limit would always reach one. It would be better to treat as a non-mathematical world if it is a world without one and two.