Could Mathematics Prove Something to 100% Certainty?

[Tcl70]

Hi there,

I have a question that may seem stupid do to my lack of proper education

I hear often the following term , "in science nothing can be 100% proven"
and recently a heard the something like "in science nothing can be 100% proven not even in math"

so this kinda got me confused because of this quote "Mathematical proofs are absolute proofs"

Q: can you prove something to 100% in math or only very close to 100% ?

PS: sorry for my poor English and my dumb *** question :-/

 

[HallsofIvy]

First, I think you must distinguish between "science", as used here, and "mathematics". The science referred to here is "experimental science" where everything is based on observation and experiment with "inductive proof" following from observation and experiment.. Since we are not capable of measuring things "exactly", any measurement is approximate and any experiment approximate. We cannot "prove" things beyond any possible doubt in science.

Mathematics, on the other hand, uses "deductive proof", not "inductive proof". Deductive proof starts with given axioms or postulates that essentially "define" the type of mathematics we are dealing with. Assuming that the proofs are valid logically, then they are 100% correct- they prove conclusively that the conclusion follows from the hypotheses. Of course, that says nothing about whether the hypotheses themselves are "true" or not. That is not the object of mathematics.

You English is excellent- far better than my (put just about whatever language you wish here!). And your question is far from dumb!

 

[AC130Nav]

Hi there,

I have a question that may seem stupid do to my lack of proper education

I hear often the following term , "in science nothing can be 100% proven"
and recently a heard the something like "in science nothing can be 100% proven not even in math"

so this kinda got me confused because of this quote "Mathematical proofs are absolute proofs"

Q: can you prove something to 100% in math or only very close to 100% ?

PS: sorry for my poor English and my dumb *** question :-/

You can prove something 100% in math because math is based on assumptions including its rules. Given the assumptions you can prove with certainty what comes from those assumptions. You don't have to prove they apply to the real world or that part of the real world you might be studying as a scientist.

 

[hgfalling]

Another way to think about is that mathematics is just tautologically true. We begin with axioms, and reason from those axioms to theorems. Those theorems are true, but they are true in a sense that is both more trivial and more significant than things in physical sciences. It is more trivial because in mathematics we are unconstrained by observations of the real world; mathematical objects are constructed and need not conform to anything at all. It is more significant though, because the theorems of mathematics aren't subject to falsification; that is, if we prove a theorem, it is true in a way that the theories of experimental science can never be.

Now applying mathematics to real-world problems involves some process of modeling the real-world situation in some way as if it were some mathematical construct. But if the results of the model do not match observation, it is the model that is wrong, not the mathematics.

 

[JonF]

It depends on what you mean by “100% prove”.

Math in one sense is certain, since all of its truths are “hypothetical” in that they are an implication of the axioms and logic. A mathematical proof/truth essentially says - if certain rules (usually something like ZFC and a logic system) are applied correctly to certain axioms are this truth follows. Notice this interpretation doesn’t mention human effort in mathematics at all. It is appealing to “mathematical truths” in the pure sense – that they exist independent of man.

In another sense it isn’t 100%. It’s possible there is an error in a proof that no one has noticed yet, essentially the rules were applied incorrectly and even after hundreds and hundreds of mathematicians reviewed it no one realized it. The odds of this are slim, but not nonexistent. I personally (but I'm sure someone more knowledgeable here can point one out) don’t know of a single rigorous proof that was widely accepted in the community for a substantial period of time that later was proven faulty. Notice this interpretation appeals to what man can know and do with mathematics, it is not a statement about the actual axiom system.

 

[micromass]

I agree with JonF. I don't believe proofs in mathematics are 100% correct, since those proofs are made up by men, and a human is fallible Take Wiles' proof of Fermats last theorem. It has hunderds of pages and only a few people can really understand it. Well, I wouldn't call that a 100%% proof, even tho it is very likely correct.

There are however some attempts to formalize all proofs in mathematics and letting it be checked by a computer. See e.g. http://us.metamath.org/mpegif/mmset.html . There are many prooifs (including Hahn-Banach and other monsters) which have been check by metamath. But then again, I wouldn't call that 100% since a computer is fallible to...

And then there's Godels incompleteness theorem. It may very well be that every proof is 100% true, but it might not mean anything, because mathematics is inconsistent. It is my personal conviction that ZFC is consistent, but nobody may really know for sure...

 

[arildno]

To make an analogy to another area:

Set up some situation with the chess pieces on the chess board.

You may then ask:

Is this a situation that CAN occur, according to the rules of chess?

And, if it is, can it be accomplished in, for example, 12 moves?

These are statements that we intuitively think ought to be provable, or disprovable, solely by means of the rules at our disposal.

In many ways, mathematical proofs seek also to prove whether a particular situation, or relation can be valid, within the set of axioms (base rules), we have chosen.

 

[jeffasinger]

To expand on what micromass is saying about the incompleteness theorem, in some sense, it is impossible to prove that mathematics is consistent. Where consistent means that it doesn't lead to a contradiction.

However, it is pretty widely believed that ZFC is consistent, and if that were the case, then there is such a thing as 100% proved things in mathematics ala the computer proof checking, however there have been cases where mathematicians believed something to be true, and thought that they had a proof for it, but were making a hidden assumption, or something of that nature, and it was later found that there proof was not satisfactory.

 

[JG89]

I think the four colour theorem was widely accepted at one period of time even though the proof was false (this was corrected and there is a correct proof now)

 

[jeffasinger]

JG89 - I believe that is correct. I think the historical proof is only satisfactory for the 5 color theorem, and that the only proof of the 4 color theorem is actually computer based, and not really accessible to a human. In my Graph Theory textbook, it said that the first correct proof was 741 pages long, not very easy for one person to follow.

A more extreme example of a proof so long that it's likely that no single person has gone through all of it is the Classification of Finite Simple Groups.

I think that many mathematicians used various forms of the Fundamental Theorem of Algebra before it was proved, in fact, I think Euler gave an incorrect proof of it at some point.

 

[Petek]

This is probably an incredibly naive question, but is it possible, in theory, to prove that a proof is valid? I'm thinking along these lines: Let's say that we have a theorem, call it Theorem 1, and a supposed proof for it. We now wish to prove that our proof is valid. Thus, we have to prove Theorem 2: The proof of Theorem 1 is valid. We come up with a proof of Theorem 2, but now we need to show that it is valid, leading us to Theorem 3: The proof of Theorem 2 is valid, and so on. How do we avoid an infinite regression of proofs?

 

[micromass]

That's actually a very good point you bring up, Petek. I don't think it is possible to prove that a proof is correct without going in an infinite regression. It's always upon the humans to show a proof is correct, and humans are fallible...