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What does it mean to prove something? I am constantly told that something is defined to work in some way (take as an example the truth table of p implies q) and that you don't need to prove it works right. But the problem is that I don't see the point in working with something which might turn out to be wrong. I am expected to take the risk?

Then proving something is just more or less about remembering all the arbitrary facts of a certain system and using this facts to explain something which happens inside of this system? Or should I assume it all doesn't matter, as long as I'll stay consequent then I'll be able to correct whatever facts are wrong automatically?

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    It would help if you gave specific examples of what you are "constantly" being told. Some things are just definitions, some things need proof based on those definitions and assumed axioms. – Thomas Andrews Oct 04 '15 at 15:03
  • Take as an example the truth table of p implies q. –  Oct 04 '15 at 15:04
  • Don't put it comments, put it in he question. – Thomas Andrews Oct 04 '15 at 15:05
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    You're using words as part of the English language. Presumably you are aware that they have agreed-upon meanings. You believe the definitions of words don't need to be proved to work right. But how do you know they won't turn out to be wrong? Why take the risk of communicating at all? – anon Oct 04 '15 at 15:10
  • I understand that you're implying that mathematics is just another way of getting your message across. –  Oct 04 '15 at 15:22

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You can define whatever you want. I say that an integer is floopy if it is even, prime and greater than 3. My definition is perfectly adequate, and I don't need to prove anything about it.

However, if I do just a bit of work, I can show that no integer is floopy. My definition is still absolutely fine; it's just that nothing satisfies it.

In that sense, I don't need to prove my definition to be true - what would that even mean? - but I should really justify that it's useful. (In this case, I can't justify it, because it describes non-existent objects.)

"Proving something is just about remembering the arbitrary facts of a system and using them to explain things inside the system" is basically spot on, yes. The art of maths is a) in finding the right system, b) spotting the facts inside the system, and c) finding the right way to combine what we know about the system to prove the facts we spot. We have settled on a certain system which seems to make sense and can be used to model reality in various useful ways - the integers, for instance, capture the notion of "number of objects" from the real world, so we study that system.

Patrick Stevens
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    So definitions are all alright, but once you want someone to agree with you have to prove something with definitions everyone can understand such that they're likely to agree with you? –  Oct 04 '15 at 15:24
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    Yeah, exactly. It's your job to convince me to pay attention to your definition. – Patrick Stevens Oct 04 '15 at 15:26
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    So math is basically what would happen to our natural language if you'd make it a lot more strict and easily calculable. And proving something is then just convincing someone else to something you want, like they do in newspapers? Sorry about all these questions, but there is so much misconception I have developed about math over the years... –  Oct 04 '15 at 15:37
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    Yes. We have developed standardised ways to make our arguments more convincing - ways to make mathematicians more confident in what we're saying - so we use those. – Patrick Stevens Oct 04 '15 at 15:38
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    I see. Thanks so much for all your answers. –  Oct 04 '15 at 15:40
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You can see :

By `argument' we mean, roughly, a chain of reasoning in support of a certain conclusion. So we must distinguish arguments from mere disagreements and disputes.

And page 37 :

When a chain of argument leads from initial premisses to a final conclusion via intermediate inferential steps, each one of which is clearly valid, then we will say that the argument constitutes a proof of the conclusion from those premisses.

This pattern is the core of mathematics : the conclusions derived by valid arguments from the axioms assumed as premises are the theorems of the mathematical theories "identified" by the said axioms.

Mauro ALLEGRANZA
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  • But how are you sure that the thing which seems valid to you today will seem valid to you tomorrow and all the time to follow? There is no divine mind which tells us that we are interpreting our own axioms correctly. How can we be so sure that everything which has been proven until now is valid? Do you understand what I mean? –  Oct 04 '15 at 15:13
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    This is the math SE, not the philosophy SE. – anon Oct 04 '15 at 15:19
  • @Matt - you cannot. If you think that math is an "activity" not worth to be "practiced", there are planty of other beautiful things to do : poetry, sport, music, philosophy, tracking,... In the human world there are no absolute certainties, neither in mathematics. – Mauro ALLEGRANZA Oct 04 '15 at 15:19