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Well, it’s not just general proof by contradiction. In constructive logic, deriving a contradiction from a positive statement to prove the negative is allowed, however a double negative does not get you the positive. But also, you lose excluded-middle arguments, where you derive a third statement from both a positive and its negative.

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bro tf is the title

So basically, classical proofs are what you might learn in a discrete math textbook. Constructivist proofs are the intuitive ones where you construct such an object, and make more sense if you learnt coding before you learnt math.

the backing track is too loud and has to many sounds in the same registers as your voice :( cool video though!

Let x=9 and y=10. Then x+y/2=14, which is not a number between x and y

A constructivist and an intuitionist walks into a bar. The bartender: "This must be a joke".

what´s the music good for?

This video is so misleading. Firstly, this is a video comparing constructivist vs non-constructivist. No "classical" mathematician would prove that numbers exist in between using squeeze theorem. That proof is just forcing non-constructivism and is a pretty bad example. See the proof of the fact that irrational power of irrational numbers can be rational. In that case, a non-constructivist proof is extremely simple. Also, this video makes it seem like mathematicians just avoid constructivist math because it's difficult and hard to teach when in reality, the main problem with it is that it cannot prove a significant portion of mathematics. When I say cannot, I don't mean it is difficult, I mean it literally cannot. This was completely ignored.

I need some explanation. If we select x = 3 and y = 4 then x + y/2 is going to be 5, which is not really in between 3 and 4? Or is it actually (x + y)/2 ?

never been here before

I'd remove the "the" in the title.

isn't the third question in the quiz a group though? it's closed since it's a cycle, it is associative, 3 is the neutral element (having 3 elements in the loop, adding 3 to n has no effect) and thus each element has an inverse (3, itself, and 1 and 2 are inverses of each other)

7:24 Cries in PEMDAS

This is a good introduction, if a little light. I didn't know that Bauer's "five stages" paper was a response! Note that our construction of the mean still proceeds by axioms: x < y, so x + x < x + y < y + y by adding x and y to both sides of two copies and pasting; then, divide everything by 2 to show x < (x+y)/2 < y. We do the same steps in the proof as in the algorithm; "add x, add y, divide by 2". This is an instance of the Curry-Howard correspondence!

Either way, its fun

5:38 "ma-zo-cheest" I'm guessing you meant to say ma-zo-keest ....

the answer is simple it is impossible to construct a machine hat can simulate it's self faster then real time.this makes sense because with every step of the algorithm it has to simulate multiple steps.and because the simulator is part of the universe it has to simulate itself so it couldn't simulate the universe faster then real time.

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A big problem with the Law of Excluded Middle in mathematics is that it is uninformative. Let's say my thesis is: "either 23 divided by 45 plus 86 is equal to 286 or it isn't equal to 286." Within classical logic, this is a tautology and a perfectly valid conclusion. However, what it doesn't tell us is whether this equation does - or does not - equal 286.

I forgot to do noise suppression, will re-upload in a bit.. maybe

Great video! Underrated content

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I read the question as half as hot and got -8.8C. I didn't understand you meant something else. Im not even sure what you mean now. Choosing room temperature as a midway point is arbitrary and isnt very useful. Why choose room temperature and not some other point say 100°C? I converted Celsius to Fahrenheit, divided by 2, then converted back to Celsius. 0°C =32° -8.8°... = 16°

Or you could just not watch porn..

I think it's potentially misleading to depict constructivism as a subset of classical mathematics rather than vice versa. In terms of model theory, it's more accurate to say that classical logic is a model of constructivist logic rather than the other way around, since constructivist logic is more general. That is, classical logic would never reject any constructivist proof, but constructivist logic may reject a classical proof; truths of classical logic are either true or undecidable in constructivist logic, but all truths of constructivist logic are true in classical logic.

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Removing Lem doesn't make it less, it makes it more. Constructive mathematics is a superset of mathematics.

LMAO

Halfway through, but it seems pointless and that there’s no reason we would ever want to get rid of the law of the excluded middle.

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this is so cool! where is it?

Unnecessary music very annoying and distracting - I'm outta here.

Actually is a common mistake to believe that in constructive (intuitionist) mathematics you can't do proofs by contradiction at all. The law of excluded middle still there in a softer form: a statement is not either true or false, but it's proved true or not proved true. If you have a proposition A that leads to a contradiction, then you have proved "Not A", even in constructive math. No problem in that. What you can't do is having a proposition "NOT A", that leads to a contradiction and therefore deduce A. This is forbidden in constructive math, because it allows the "theological proofs of existence", without show the object that is claimed to exist. That's because is not " the law of excluded middle" that is not valid, but the counternegative: "not not A", doesn't mean A. In constructive math A is true means that, using axioms, you proved A. "Not A" means that, using axioms, you proved that A can't be proved (and It couldn't be proven even if you add tailor-made axioms that do not lead to contradictions). "Not not A" doesn't mean "A" in constructive mathematics, but it means that, using axioms, you can prove that is impossible to prove that A can't be proved (even by adding tailor made axioms, without leading to a contradiction). It seems a twist tongue, isn't it? The fact that the law of excluded middle isn't valid at all in constructive math is a very common misconception that makes most people rise the eyebrow and refute to use constructive mathematics. It's not about the LEM, it's about what you mean by "true". In constructive math it means "provable", while in classical math it is thought that a statement can be true or false, but maybe not provable with the current set of axioms. That is the problem with classical logic: that metaphisical concept of truth that can trascend the system of axioms you are in.

no bgm pls

I decline to hear this as the music is too loud!~

"Background" music ruined the video

Can't believe tovy fox create maths

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I initially thought that classical logic is a subset of constructive logic, since constructive logic has less inference rules than the classical, thus has the potential to add more non-classical inference rules. But everything accepted by constructive logic also accepted by classical logic, so constructive logic is a subset of classical logic like what this video said.

It's not that constructive mathematicians don't believe any statement is either true or false, it's rather that they just don't care. Constructive logic, as its name suggests, is not about truthness but about what you *can* construct (proofs). For instance, the "there exists" symbol in classical maths only means that there theoretically exists such an object but with no guarantee at all that you can find it. While in constructive maths, the only way you have to prove a "there exists" statement is to actually exhibit such an object. This is the same reasoning for the LEM. As a matter of fact, everything you've proven with constructive maths also stands in classical maths, but not the other way around. Having a constructive proof with you is more powerful as it just says more, so that's why you should fallback on classical proofs only when mandatory and when you only care about truth.

idk what the point of the video is but i enjoyed it also nice editing

Nice material. But instead of wasting your time with that cartoon childish nonsense, focus on your math material. You will be way more prolific.

cherry bomb in a random maths video?!??? based .,.,,,