This argument is seemingly related to Gregory Chaitin's views on noncomputability and nondefinability. Chaitin observes that almost all real numbers cannot be referred to, or singled out, by any mathematical method available to us -- for example because definable numbers using a language or notation must have the cardinality of the natural numbers, but we know from Cantor that the cardinality of the reals should be larger.

Chaitin simply thought this was an impressive fact about the reals and the limitations of mathematics -- a way in which mathematics contains randomness and that many or most facts are "true for no particular reason". This author instead seems to conclude for a related reason that the reals don't exist because we have (and could have) no usable technique to distinguish most real numbers from one another. His complaint in this video is a Chaitin-like observation that we have no way to distinguish real number A from real number B in a finite amount of time or with a finite amount of reasoning or information, and an un-Chaitin-like conclusion that maybe we then have no reason to believe that these numbers exist and are distinct from each other.

Edit: and he emphasizes later that if we believe in the reals, numbers must exist that we can't actually do arithmetic with (which I would suggest is sometimes for the Chaitinesque reason that we can't name or define them, or other times for the weaker Chaitinesque reason that we can't calculate their values), so he seems to ask what good such numbers are to us or what reason we could have to believe that they are real.

(For anyone else reading this who might be misled, note that OP's position is generally considered to be "crank" mathematics)

For anyone who might be misled, here are some links that people can use to judge for themselves how "crank" someone is:

Professor Norman Wildberger's homepage:

http://web.maths.unsw.edu.au/~norman/

...a paper by Wildberger tackling some of the issues in this thread:

Set Theory: Should You Believe?

http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf

Some interesting works by Gregory Chaitin

Meta Math! The Quest for Omega

https://arxiv.org/abs/math/0404335

Exploring Randomness

https://www.cs.auckland.ac.nz/~chaitin/ait/

How Real are Real Numbers?

https://arxiv.org/abs/math/0411418

People who are interested in ultrafinitism would also want to check out Doron Zeilberger

http://www.math.rutgers.edu/~zeilberg/

and

Edward Nelson

https://web.math.princeton.edu/~nelson/

It is definitely not a standard or mainstream view, but it could be a flavor of https://en.wikipedia.org/wiki/Finitism which has been defended by a very small but not infinitesimal :-) number of professional mathematicians and which isn't a logically inconsistent position.

Personally, I am an ultrafinitist.

By that I mean that I believe that physical systems can be completely described by constructive mathematics based on intuitionistic logic[2] operating on computable reals[3]. I believe that any other kind of mathematics, e.g. classical logic with axiom of choice can create unphysical models.

That being said, I don't object to classical logic as a purely abstract concept. Everything proved in ZFC is certainly true in ZFC! And I don't think any finitist will contest that.

[1] https://en.wikipedia.org/wiki/Ultrafinitism

[2] https://en.wikipedia.org/wiki/Intuitionistic_logic

[3] https://en.wikipedia.org/wiki/Computable_number

You got as much rigor as you gave. Disproving every wacky position with a careful analysis is impossible, nobody has that much time.

Indeed, there are an almost infinite number of crank theories because they don't require logic, evidence, or proof - just belief. One crank can churn out a hundred nonsense theories in the time it takes someone to validate one scientific idea or mathematical proof.

This has a corollary in the startup world: everyone has an idea, what matters is execution.

Uhhhh...no. Take a look at Dedekind cuts.

The linked video is largely a critique of Dedekind cuts, arguing that they don't in general let us recognize, distinguish, or perform arithmetic on most real numbers. (Almost all of the informational input to a Dedekind cut for a randomly chosen real couldn't be written, remembered, or specified in any way by a human being.)

I think the presenter in the video is trying to justify a kind of finitist attitude based on the inaccessibility and unspecifiability of reals-in-general to us. This could also be advocating a position something like

https://en.wikipedia.org/wiki/Computable_number#Can_computab...

Edit: or perhaps https://en.wikipedia.org/wiki/Constructive_analysis (I didn't watch enough to understand exactly what alternative he proposes)

> Almost all of the informational input to a Dedekind cut for a randomly chosen real couldn't be written, remembered, or specified in any way by a human being.

Well, yes. The reals are uncountable.