What is a number?

Joel David Hamkins, Professor of Logic, Oxford University

Lecture 1. Numbers

Numbers are perhaps the essential mathematical idea, but what are numbers? There are many kinds of numbers—natural numbers, integers, rational numbers, real numbers, complex numbers, hyperreal numbers, surreal numbers, ordinal numbers, and more—and these number systems provide a fruitful background for classical arguments on incommensurability and transcendentality, while setting the stage for discussions of platonism, logicism, the nature of abstraction, the significance of categoricity, and structuralism.

[DrThalesAlexandre]

Thank you for opening up these lectures to the general public. It is a privilege being able to watch them.

[randylayhee]

Philosophy major who became a software engineer here. Really appreciate you opening this up on youtube. It funny how the more I learn about computer science the more I become interested in philosophy again.

[Israel..]

University lectures being open shouldn't require special circumstances of the pandemic. Openness is a force that accelerates growth of scientific knowledge.

[joeldavidhamkins]

Errata:

At 6:52, I should have said that the 3/6=1/2 example would violate Leibniz's principle on the indiscernibility of identicals, rather than the identity of indiscernibles, which comes up later.

At 20:28, I should have said that we reduce one equivalence *relation* to another, rather than an equivalence class.

At 47:16, I should have said that the complex numbers are characterized as the unique algebraically closed field *of characteristic zero* and size continuum.

[romanbesel]

So cool that you made this open to the public! Im just reading Linnebo and this series serves me as a good complementary (probably have to get your book as well). Thanks for spending the time and effort, Professor!

[erincarmody]

Oh Joel, you are so wonderful. Sorry I missed this lecture. I thought it was 11am NYC time, but I will be sure to be there next week.

[DannyYee]

It's amazing how many different ways the auto-captioning manages to render "Frege" and "Dedekind" - it's almost as if it isn't allowed to use the same rendering twice.

[BartoszMilewski]

In category theory, a natural number is a morphism from the _terminal_ object, rather than the _initial_ object.

[berkeleycodingacademy]

I am a long time math teacher, a lover of math, and a lover of the philosophy of math. Thank you so much for publishing these lectures! (Playlists would be helpful.)

[exantefacto]

Taking natural numbers starting from 0 provides intuitively and formally a smooth uniformity for many mathematical and extra-mathematical contexts. For 1, 2, 3, ..., non-zero integers, Z+, is quite satisfactory.

[cjfthistle]

Came for the maths, stayed for the outstanding fashion sense.

[Media_Mavrick]

Thank you for this information!
I have been lacking such knowledge all my life
You truly are an inspiration to the future of this world.

[Adventures_of_Marshmallow]

To me it's like this: There's nothing, there's everything and then there is you. The realization that you are neither nothing nor everything gives you one.

Numbers, like language are the result of the capacity to systematically isolate and give identity to existentials through the comparative of nothing and everything and is filtered through self. "this is not that, but it's also not me." .

[nothingposted]

I fell asleep listening to this, woke up and, for the first time in my life, picked up pen and paper and made math ecuations for fun

[primefactor]

prof ... please make lectures on set theory you are awsome

[lunavicta]

On the topic of synthesia, I remember reading about a man who could identify primes up to a certain (high) number based on a characteristic "roundness". If he was doing that from an early age, that suggests to me that the concept of a prime number has some physical existence in the structure of the human brain.

[cookiecrumbles]

“The number 1 is the class of all classes such that there is an element having the property that membership in the class is interdeducible with being equal to that element.” — Haskell Curry book on logic

[radientbeing]

Does Conway's game theory derived infinite surreal numbers, imply mathematical pluralism? What do you think of Woodin's Ultimate L approach?

[roberthvistendahl]

Thank you for explaining a lot to me, I loved it!! I bet it's no coincidence that you're a great bloke and so is my uncle who went to Oxford, I went to Sydney University to study maths which was the time of my life, you make me wonder whethere there are....infinity or 2^aleph0 ways to coincide with a students current understanding of the material.

[flux]

The jokes from the book are quite funny. Good stuff!