Crisis in the Foundation of Mathematics | Infinite Series

What if the foundation that all of mathematics is built upon isn't as firm as we thought it was?

Note: The natural numbers sometimes include zero and sometimes don't -- it depends on how you define it. Within logic, zero is always included as a natural number.

Correction - The image shown at 8:15 is of Netwon's Principia and not Russell and Whitehead's Principia Mathematica as was intended.

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Mathematics is cumulative -- it builds on itself. That’s part of why you take math courses in a fairly prescribed order. To learn about matrices - big blocks of numbers - and the procedure for multiplying matrices, you need to know about numbers. Matrices are defined in terms of - in other words, constructed from - more fundamental objects: numbers.

References::

Philosophy of mathematics (Selected readings) edited by Paul Benacerraf and Hilary Putnam

Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow

Thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!

And thanks to Nicholas Rose and Mauricio Pacheco who are supporting us at the Lemma level!

[eleandrocustodio]

this is one of the best PBS's shows. They should hire her back!!

[shakesmctremens]

I feel very comfortable not having to write out a few hundred pages of justification whenever I need to add 1 and 1.

[jmzorko]

This is still one of my favorite episodes, dealing with one of my favorite areas of math / philosophy. Infinite Series, as well as other channels like 3blue1brown, go a long way towards making complex math a lot simpler for peeps like me to follow. This / the world completely fascinates me :-)

[max]

Astronaut #1: Wait, it's all just turtles?
Astronaut #2: Always has been.

[LucidStew]

My crisis in math is that I've covered vectors in 4 different classes now, and I still don't know any more about them than did after the 1st class.

[acuriousmind]

That's actually Newton's Principia in the graphics.

[theexnay]

This is like the problem of how do you define a word when its definition must contain another word and as such eventually you must do the nonsensical thing of using a word to define a piece of what defines it. Despite this we can communicate clearly and that just seems magical to me, and similarly maths seems solid and logical and clear despite it being based on axioms themselves with limited foundation. I think this type of thinking is just a wonderful part of the human condition as weird as it is to understand something you can't define without using itself.

[SpaceLordof]

Your pic of Russell's book is of Newton's Principia, which is 200 years older...

[RN]

@9:00 I like how we just cruise past the implications of Gödel.

[benji]

Frege: finally I can put a foundation to math
Russell: think again bro
Frege: woah you log-blocked me bro

[senorgooba]

The title scared me. I was expecting a Gödel's incompleteness theorem type discovery but more deadly.

[joaoenes]

Just a minor and natural mistake, the Dedekind cut for the square root of two is wrong. The sets should be {x in Q | x^2 < 2 or x < 0} and {x in Q | x^2 >= 2 and x >= 0}.

[LloydDT]

8:16 Wrong Principia... The editors must be out to lunch again.

[musikinspace]

I swear when you said "What grounds mathematics?" I thought "It's turtles all the way down" and then you said it. I must be psychic

[kansascityshuffle]

Admittedly I keep videos like this to help me fall asleep. Have to wake up early because I got a Cs and Ds in high school.

[offtheball]

For anyone looking for the last link:


It may be added to the description soon, but might save you typing it out.

[TrevorKafka]

I'd love if you could dig deeper into the ZF axioms and provide an example of how the axioms could be used to derive a theorem in classical mathematics.

[hartosaarinen]

A video on foundations without mentioning Cantor, that’s something

[daca]

Since subscribing to your Chanel, I find myself picuring you explaining thing to me when I study maths for an examn or had to proove soume engineering theory mathematicly. You restored my love for maths! Thank you for that!

[balthazarbeutelwolf]

Dedekind cuts are usually (e.g. in Landau's book) just defined for positive numbers, and then negative reals are defined on top of that. The problem is that the lower set in the video contained numbers like -7, and (-7)^2<2 happens to be false. Alternatively, you can require that the condition defining the sets has to give you ideals - in which case the simple x^2<2 cannot be used as a cut.