Intuituionism I: Brouwer's CONSTRUCTIVISM ⟨06,05⟩-optional

[@salman-11924]

This is the broadest, simplest, and most eloquent explanation of constructivist mathematics I've come across thus far. Very well done sir!

[@LogicwithBo]

Thank you!

[@elizabethalexandrov3289]

Looking forward to part 2!

[@alaaalyaseri3522]

Same!

[@nolestock4291]

I had no idea there were concrete links to philosophy into mathematics

[@gogigaga1677]

BIG BIG BIGGGGGGG THANK YOUUU!!! GREAT WAY TO START 2023 I LOVE HOW YOU EXPLAINED IT WITH SUCH SIMPLICITY

[@LogicwithBo]

Glad it helped! :)

[@philosophyversuslogic]

Your videos are truly inspiring, and magnificently informative. I am fond of watching lectures like that. What can I say - this is brilliant! Thank you! Kharkiv (Ukraine) watches you!

[@LogicwithBo]

Thank you! I am so happy you're enjoying them :)

[@vilia5482]

1:11 How could this be the case when mathematics is a timeless subject? i.e. studies things which structurally and in what defines them remain the same no matter how much time would pass

[@LogicwithBo]

Intuitionism rejects the idea that there are objects of mathematics that exist in some timeless realm. So it would reject the Platonist conception of these things from the get-go. I hope that answers your question!

[@vilia5482]

@@LogicwithBo Would they say that we just didn't know that some things would have been completely changed in how mathematics works without us noticing because you can only access it through your intuition, but that simply is not the case, at least with modern mathematics you could not pass for a mathematician if you wouldn't have learned about some abstract mathematical concepts and if they can change over time what happens to the theorems you prove with or about them? I would claim that coherence which is what any conceptualization ultimately depend on is independent from time what would you say?

[@littlefishbigmountain]

@@vilia5482 Just food for thought, everyone knows 1+1=2, but in binary 1+1=10, but they express the same underlying idea that this thing and that thing are together two things. Then there’s addition modx, so in addition mod 2 (using 0 and 1) 1+1=0, which is describing a fundamentally different thing then numbers of objects, so what does 1+1=? It depends. Although, all these possible mathematical structures and systems we can make, while they legitimately differ, do retain their structure and identity over time, and hypothetically could’ve been discovered by anyone at any time if the circumstances were right. Is having a 10,000 page proof that all finite simple groups are known an invention or a discovery? I’m also inclined towards the latter.

[@MathCuriousity]

May I ask: what does it mean to talk about “non axiomatic math and logic systems” ? I saw this term somewhere. Do these systems still make assumptions but we just don’t technically call them axioms?

[@LogicwithBo]

Do you recall the source? My guess is that this is meant to contrast them with axiomatic systems like that of Russell and Whitehead.

[@MathCuriousity]

@@LogicwithBo to be honest it’s over my head and I just started learning about logic. But I stumbled upon “non-axiomatic” systems of logic and math and so am wondering how it’s possible to have systems that are valid but make zero assumptions about anything? (Non-axiomatic meaning no axioms and not even rules of inference)! So can you help me understand how that’s possible? Finally, am I wrong to think that constructivism and intuitionism do make assumptions just not in the form of axioms or rules of inference?

[@LogicwithBo]

​@@MathCuriousity hmm without the context I can't say, but certainly many older systems, like Aristotelian logic, are not fully and completely built up from axioms. Like there is no set of axioms that lies at the root of the Prior Analytics, Sophistical Refutations, and Topics. These are mostly around arguments as they happen in day-to-day language and reasoning. But they are not non-axiomatic in the strong sense of lacking no basis. They just don't have one single set of axioms at their root. Intuitionism is a type of constructivism. And I think it's fair to say that any approach to mathematics and the philosophy of mathematics makes certain assumptions, be it classical/platonist or constructivist. I hope this helps!

[@oneofvalts]

such a nice video. so articulate.

[@LogicwithBo]

Thank you! :)

[@woosix7735]

It seams absolutly crazy that the inventor of constructivism also invented topology. you know, the feild with proof by contradiction absolutly EVERYWHERE

[@littlefishbigmountain]

Typically the people that come up with revolutionary ideas don’t think as highly of them as everyone else seems to (once they recognize their value). An extreme such example is Paul Dirac, and there are many many others

[@woosix7735]

@@littlefishbigmountain Ok... I don't get what this has to do with my comment, would you mind elaborating?

[@littlefishbigmountain]

@@woosix7735 Just because someone revolutionizes an alternative doesn’t mean they’re 100% sold out on it to the exclusion of the methods one’s studied all one’s life

[@patpearce8221]

Chatgpt could of told me the same things