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Is there a bottom line to what should be proven?

This might be a bit long, but certainly of interest to any mathematician!
TL;DR JUST ANSWER TITLE

Any good mathematician would tell you that proof is important, and indeed it is, I myself as an AS level FM student has myself either proven, or viewed the proof to all the theorems which I've used or covered (from why the volume of a sphere is 43πr3 \frac{4}{3}\pi r^3 to the fundamental theorem of calculus to explain exactly why the process opposite to differentiation gives the area under the curve).

But my question is, is there a bottom line in which proving things which precede it is unnecessary or a bit too much?

For example let me give you a question: Consider the equation x+2 = 4, prove that there is only one solution to this equation. Is something like this too much to ask? It may seem obvious but to a rigorous mathematician, intuition is not enough: everything must be proved. It is provable and it is easy to, see if anyone can do it before opening the spoiler.

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So where, if it exists, is the bottom line to what should and shouldn't be proved? As if there is a bottom line we would spend absolutely ages proving the most simply frustating things (which I somewhat agree with doing), and if there isn't a bottom line then maths wouldn't be maths: we could just assume things without justification and that's not good :wink:.

My opinions to this below :smile:

I'm curious to other people's thoughts to this so post away without hesitation, cheers! :colondollar:

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(edited 10 years ago)
Reply 1
Mathematics is axiomatic, so at the bottom of mathematics are little building blocks known as axioms. Axioms are like the rules of a game, you set them up and you see where the logical trail of ideas leads you to. You don't prove these axioms, they are just set up so that you can build mathematics above it. A popular view of axioms is that of self-evident truths, which some people may find disturbing as everything else in mathematics is proven yet these things are just "obvious", so seeing them as part of the definition of your mathematical system might make axioms seem slightly more comfortable.
As the above poster says, the "true" bottom lines are sets of axioms ("non-logical" axioms, I believe). In mathematics, we "assume" these things to be true, and then derive the rest of mathematics from there... so mathematical statements can only be said to be "true" in the context of said axioms. We prove all our statements, but we have begun from the assumption of said axioms and hence the truth value of the statements is dependent on them. Some axioms may be "self-evident" truths (if there can be such a thing), but some are simply defining properties e.g. the existence of a null vector in the definition of vector spaces.

If we started from a completely different set of axioms (assuming there exists another such set from which it is actually possible to derive anything), I see no reason in principle that we couldn't derive completely new variations of mathematics (whether or not these variations would reflect current fields is something I wouldn't want to comment on). Some clever person somewhere may have somehow proven that this isn't true, but if so I don't know about it (though that isn't saying much).

But generally speaking, the level of proof needed is context dependent. If the audience for whom you are writing accepts the validity of a statement, there is typically no need to prove it. For example, in an exam your lecturer will typically have given you a bunch of theorems (whether explicitly or implicitly) which you are free to use without proof and so this is "where the buck stops", as it were, in this instance.

I think a schooling system in which every statement was proven from very first principles would be essentially impossible to implement... there is a reason pure mathematics is not usually introduced until university.
Reply 3
To answer your question in a nutshell, the bottom line is decided by people, who think up some rules that result in useful mathematical results.

You should do maths at uni - in your first year you would study this kind of thing. Essentially you start off with some definitions of how addition, multiplication etc work then you prove things from that. You would initially prove things that are really trivial (given any non-empty set of positive integers, there will be a smallest element of that set) but then move on to more complicated things like why the formulas for differentiation/integration work. It's only occasionally when you would be given some formula without justification.

Later on, if you take modules in set theory, you can study a way of defining numbers based on sets, which in turn are defined by certain axioms (rules thought up by people for how sets have to work). Have a look at http://en.wikipedia.org/wiki/Set_%28mathematics%29 for more about what sets are.

What you can sometimes do is investigate what happens if you try to change these axioms or definitions. For instance, suppose you take just the integers and define in addition and subtraction but not multiplication or division. Then include the rule 12=0. This gives you a mathematical model for dealing with numbers on a clock face. For example, 14=12+2=0+2=2.
Reply 4
Cheers for the input of you guys, you all bring up very interesting points.

Consider this: to the view that everything can and should be proven, I'd like to bring attention to the 5th axiom of Euclidean geometry: the parallel postulate. It states that

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Interestingly, this cannot be proven (from what I've read, correct me if I'm wrong) and hence there does exist a bottom line, although one could argue that since it's an axiom it of course can't be proven. But what if I make a new maths rule, call it an axiom so it's automatically true, and it turns out to be wrong and everything falls down after that?

Also do you guys think that (morally) we have become a bit lazy, using words like 'trivial', 'obvious', 'I have a marvellous proof, which this margin is too tiny to contain!' to obscure the mathematical beauty, and necessity, of proof?

On another note I am definitely planning on doing maths at uni, hopefully Cambridge :wink:, to unveil the mystery behind all of this!
Original post by Omghacklol
But what if I make a new maths rule, call it an axiom so it's automatically true, and it turns out to be wrong and everything falls down after that?


Well that's the thing, how would it "turn out to be wrong"? The only way it could be "proven wrong" were if it were shown to be wrong relative to any other axioms we have assumed. Unless it were a self-contradictory statement, we would have to employ other axioms and their derivatives in order to disprove it. Which would simply mean that it was inconsistent with that which we have already assumed and not that it was objectively "wrong".

Remember we're talking of abstract concepts so it's not like we can just do some science and "compare it to observation". There is no absolute.
(edited 10 years ago)
Reply 6
Original post by Omghacklol
Cheers for the input of you guys, you all bring up very interesting points.

Consider this: to the view that everything can and should be proven, I'd like to bring attention to the 5th axiom of Euclidean geometry: the parallel postulate. It states that

Spoiler



Interestingly, this cannot be proven (from what I've read, correct me if I'm wrong) and hence there does exist a bottom line, although one could argue that since it's an axiom it of course can't be proven. But what if I make a new maths rule, call it an axiom so it's automatically true, and it turns out to be wrong and everything falls down after that?

Also do you guys think that (morally) we have become a bit lazy, using words like 'trivial', 'obvious', 'I have a marvellous proof, which this margin is too tiny to contain!' to obscure the mathematical beauty, and necessity, of proof?

On another note I am definitely planning on doing maths at uni, hopefully Cambridge :wink:, to unveil the mystery behind all of this!

Like it's been said, from me and others, axioms are just the basic rules of the game. Say you have 4 axioms A, B, C and D, then for E to be endowed with the title of being an axiom, you must show that you can't reach E using the properties A, B, C and D, otherwise E would just be a theorem.

Also if you have verified this, then the mathematical structure created from these axioms will be completely consistent so long as you stay within the boundaries of the axioms.

For example, take a Group. A Group is endowed with the properties of closure, associativity, identity and invertibility. So you must not assume every group is commutative, because that isn't one of the axioms.
Reply 7
Original post by Omghacklol
But what if I make a new maths rule, call it an axiom so it's automatically true, and it turns out to be wrong and everything falls down after that?
That kind of thing has happened in the past. One example is Russell's paradox: http://en.wikipedia.org/wiki/Russell%27s_paradox Here, the axioms of set theory were such that too broad a range of objects were allowed to be sets and this led to a contradiction.

Also do you guys think that (morally) we have become a bit lazy, using words like 'trivial', 'obvious', 'I have a marvellous proof, which this margin is too tiny to contain!' to obscure the mathematical beauty, and necessity, of proof?

One reason why this sort of language is used is that if we got bogged down in all the details for every proof then it would be harder to understand it and see what the key ideas are (believe me - there are some very hard to understand proofs out there). Of course, you do have to make sure that an argument really is trivial - Fermat's last theorem is an example of where the argument was in fact a lot more complicated.

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