A Socratic look at logical weaknesses in modern pure mathematics

[@_GrayBear_]

Another excellent talk. I was very much looking forward to it and it lived up to expectations (yet again). Kudos to those that did the filming and editing as well. I watch a lot of lectures, and this was put together very well.

[@douggwyn9656]

John Graber Good point. KZbin has many theoretical-physics videos, such as the category theorists' at Cambridge UK, that are practically unintelligible due to bad sound or camera work. And a lot of videos focus on the presenter rather than on what he is writing or pointing to. It's nice to see the task well done.

[@njwildberger]

Doug Gwyn Thanks to Daniel and Gennady for an excellent job videoing!

[@mickwilson99]

The Steve Martin of maths!

[@thomasfuhrmann]

A great summary of some of njwildberger's arguments.

[@CaspianBaska]

I highly enjoyed this.

[@michaellewis7861]

I always need to return to these lectures after learning new math for the sanity check.

[@brendawilliams8062]

I just check to see if there’s an improvement. I know I am a turtle 🐢. Those Fasr Rabbits 🐇 with everready batteries! And then we know we have Dr. Wildberger , Yes!

[@ThatAnnoyingINTJKid]

great talk, I really appreciate your descent from commonly accepted foundations I personally think infinities have their place in mathematics, but I agree that your much more structured & rigid perspective is superior in many aspects and should be investigated more seriously: I would sort of call your intended form of mathematics 'completely provable,' where as mathematics with infinities can only 'approach proof,' like limits

[@santerisatama5409]

Bishop Brett Does your 'completely provable' hypothesis survive the Gödel test? Would be interesting to see what someone with technical understanding of Gödel's proof would answer.

[@ThatAnnoyingINTJKid]

Santeri Satama I didn't mean that the subject itself was completely provable. I was intending 'completely provable' to describe what we consider a legitimate proof. like saying that the proofs are in some sense complete: which can imply the existence of limits, or some notion of compactness or order Since this is a restriction from the typical notion of proof due to our requirement that a proofs has a more structured form, this subject would be even less complete than ZFC since proofs cannot rely on utilizing infinities to reach theorems, which would likely significantly increase the amount of theorems that are unprovable. In a sense 'completely provable' is more of a description of local properties rather than global

[@santerisatama5409]

+Bishop Brett So, if 'legitimate proofs' are in some sense complete, in what sense? Jargon like ZFC, omega completeness, recursive numerability etc. is very difficult to fully comprehend and communicate meaningfully (at least for me), so I rather go back to the first lecture where natural numbers where created, by stating that 1) let there be 'stroke' called 'one' 2) that we can add 'stroke' aka 'one' to itself, and call the result 'two'. 3) and that we can iterate, to keep on adding one to what we already got. This way we get natural numbers defined as "strings of ones". Those are the stated foundations aka "axioms" of NJW number theory, and hence, basis of NJW proof theory. What I'm trying to ask and understand is, how does this foundation relate to ideas behind jargon like omega-consistent etc. that feature centrally in modern post-gödelian proof theory. And to my modest understanding, the principle of additivity (iteratively without a halting limit?!) as stated here, leads to incompleteness in terms of Gödel's proof, but I can be mistaken.

[@snatchngrab8262]

Santeri Satama A misunderstandings of Gödel's Incompleteness theorem is that when he said a consistent system cannot be prove from within the system, that this means it can't be proven. It can't be proven by itself... which we knew before Gödel made himself famous... it's called circular reasoning. This is basic logic, and logic is the system outside of maths that can prove it true.

[@njwildberger]

We will eventually need a discussion as to whether Godel's work in this direction properly belongs to mathematics or to philosophy.

[@richiedon100]

Great talk! Thanks Norman. It was a little frustrating not knowing what the vote totals were on each question that you asked the audience. Also could you elaborate briefly on "the most beautiful way of thinking about real numbers" as an infinite path in the SB tree? Doesn't that imply real numbers (and infinite paths) exist?

[@njwildberger]

richiedon100 I will be talking about Continued Fractions later on: this is another way to try to conjure `real numbers' into existence. It turns out that `infinite continued fractions' are pretty much the same thing as `infinite paths down the SB tree. You might like to think for example what the sequence LRLRLR... corresponds to --assuming temporarily that it really makes sense to consider such things. The reason that this method of `constructing reals' is not used in texts is that defining the laws of arithmetic are even more computationally complicated than for the other more usual approaches.

[@richiedon100]

njwildberger Thanks for the prompt reply. Looking forward to the future videos in this series.

[@santerisatama5409]

njwildberger SB tree and paths down it sounds fascinating. But I don't see the connection with reals, as they are axiomatically, or rather combinatorically constructed ("upper bound" is just fancy name for combinatorics), and combinatorics of all possible strings of a given base contains mostly non-algebraic noise that for no good reason are called "transcendentals" like algorithmic but non algebraic metanumbers pi and e. The choice between "choice" and "algorithm" is not clear cut, what you seem to mean by "choice" can be said to refer to simple combinatorical algorithm creating all the (pseudo)transcendental noise that some confuse with continuum.

[@mathspointforahsecstudents2431]

awesome video

[@MrMrkBo]

Hi Dr. Wildberger. Thanks for all your efforts to keep mathematics consistent and properly founded. I doubt you will read this, however if you do, I want to say that I am currently working my way through your math foundations series. School is back in session (I am studying microbiology at bgsu), so it may take me awhile to work my way through all of them. I have learned a lot from you, and I respect your work. Unfortunately, I think you are asking the wrong fundamental question. I believe that the question is so basic, that you may be blind to it. It is a question that ultimately defeats many of your arguments. I will not state it here (partly because I doubt you will even see my post) because I am going to watch your math foundations videos first to ensure that I am aware of your entire case before attempt to disagree. It is possible that you may address my counter in a video I haven't seen yet. Thanks for all you do.

[@santerisatama5409]

+Mark Botirius Come on, out with it. There are lots of us here, reading and thinking about fundamental questions. :)

[@brendawilliams8062]

Dumb and dumber here. 🤷?