Do you guys have any tips for mathematical proofs?
[pic obviously nrelated]
>>7934617
When in doubt, go for contradiction.
>>7934632
this
Doing nothing but referring to the definitions religiously will get you through your first proof course and probably most of Real Analysis
>>7934632
"Proofs" by contradiction are not mathematically rigorous though.
>>7934650
Please stop with this meme. You might have made the choice to work without the law of excluded middle (or might just be trying to be a contrarian) but understand that the overwhelming majority of mathematicians do not and that it is not standard practice.
Most people might prefer not to use it for aesthetic reason, but it remains a perfectly valid proof method and telling beginners otherwise is irresponsible.
Now, go be toxic somewhere else.
>>7934652
Doodle a lot, do computations, try to get a feel for what is going on.
>>7934650
>rejecting the axiom of choice
>>7934650
It is rigorous, it's just not the most insightful way to write a proof.
>>7934677
>thinking something is necessarily either true or false
>>7934632
This method makes the [math]0^0 = 1[/math] memers eternally silent.
>>7934759
Why don't people understand how conventions work ?
>>7934650
(You)
>>7934632
wtf faggot no. contradiction is last. try a direct proof first as it is the fastest. else, go for a contrapositive proof. if all else fails, use contradiction
For your first few proof courses most of the proofs you do are pretty much the same. Just focus on trying to understand the proofs they do in lecture and imitate those methods. Make sure you understand predicate logic before you attempt any proofs.
See 'some remarks on writing mathematical proofs' by John Lee. I've met him, he is a manlet but topologically equivalent to a real human.
>>7934660
I'm not that guy but teaching students to read and rely on non-constructive arguments is retarded.
Sure, producing a constructive proof is often a lot harder and requires more work but the payoff in intuition is far greater than what you get by saying
>lel, I can't give you an example of an x, I can only that if it doesn't exist then my system doesn't work and the things I want to be true are no longer true. So it must be true because otherwise I will be sad and I'll cry and cry and cry.
>>7935449
Why is it retarded ? Is it logically not sound ? If it is logically sound, then it works, period. You may choose to believe otherwise but it does work.
Math is a game of conventions and the majority of people choose to trust the excluded middle. Until they change their mind (which might happen sometime this century), I will scratch all my proofs by contradiction.
Until then, I stand by the fact that it is a perfectly valid principle and usually a quick way to get results.
>>7936095
Because one is a proof you can apply and another is a proof that (while logically valid in classical logic) is pretty much useless for anything except making other useless proofs.
A constructive proof not only tells you that a statement is true and gives you a process for creating instances of it. This is particularly useful to the novice mathematics student since they'll often want to verify or build intuition about proofs and statements by creating/examining instances and examples of said statements.
>>7937714
>contradiction proofs are useless
there's tons of things you can't prove constructively
why the fuck would you handicap yourself? for a ridiculous sense of purity?
>>7937718
There's more than one ideology in intuitionistic mathematics. The more common one in modern times is that
1) Constructive proofs should be preferred over non-constructive proofs.
2) Classical logic = Intuitinistic logic + LEM.
In other words, there are cases where LEM is false in intuitionistic logic and cases where it's true. So one should always go the extra mile and point out when we're assuming non-constructive stuff like the axiom of choice or LEM (or equivalently, double negation) in order to make a proof work. This is not just more rigorous but it also keeps is aware that without these non-constructive assumptions there may exist models where our statement is false.
3) By moving towards the more general intuitionistic mathematics we can begin to study other weirder axiomatic systems with anti-classical axioms. This isn't that different from the time period where mathematics stepped away from Euclidean geometry and began to study it as just one special type of geometry among many other non-Euclidean geometries.
The goal isn't to tie our hands behind our back, but to step back and see the bigger picture. Besides, at least some non-constructive proofs are often weird unnatural. I'd think most anyone working in mathematics has encountered at least one that didn't sit well.
>>7937752
I obviously prefer a constructive proof and will give one if it's convenient. But I'm NOT going to be pointing out whenever I use LEM and I'm not going to consider not using it. Logic without LEM is very pathologic.
>>7937762
Also, I think it's very clear when you use zorn's lemma or an equivalence that you're using it. I definitely wouldn't just use something as big as that without stating it
>>7934650
>I reject the intermediate value theorem, the post.
