What is a number? 

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.

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

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.

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

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." .

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.

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!

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.

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.)

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.

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.

Fascinating overview

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

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.

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

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

Great video series thanks! Are you going to be addressing Dialetheisms at any point in this series? Also - Can I request that the camera be perpendicular to the blackboard (pleaaase)?

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

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.

You mention that ‘what is number’ is dependent on whether it can be discerned or not by expression. So, what is expression? If I mention what is a number, would the common-sense response be: ‘one, two, three’, a cardinal, or some ordinal relation to the understanding of positionings in cardinals? Would the discern be related to language and symbolism? And the significance in the judgement from this, although, expression will fall under the fault of definitions or definiendum’s.

111001 is are tetraedric face or 4-vertex simplex face of 6-vertex simplex wich represent binary 6-digit number system.

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

I read through the chapter in the book but I actually don't quite get the closing quote by Kate Owens about "i isn't the only imaginary number ...". Can someone explain to me why this quote fits in this context?

Professor, have you considered arithmetic savants?

What does it mean by every number is a palindrome given a large enough base? I can't understand it as I can't visualise this. Can you please explain? Thank you

Great!

Professor Hamkins, Is an irrational number a number if we cannot identify or write it with a specific and precise notation? Can we even think of it, if we have no way to pinpoint it relative to other numbers which are known? Does the fact that such an irrational number cannot be specified, except as a member of a general abstract class of such "numbers," detract from its fundamental character as a number?

Would going from a vector space to an affine space - which is often described as "a vector apace that's forgotten where its origin is" - be "de-Leibnizian-ization"?

thank you love from india ...

I just bought your book, so I hope you don't give away the ending in this lecture.

Hello everyone! I suddenly realized that imagine numbers was invented approximately in the same period as capitalism and Protestantism. Also, it has some correlation in music. The madrigal as a genre of music blossomed in this period. How do you think, is there any correlation? Maybe there are any papers on this topic. Thanks=)

Love it. love numbers.

May I respectfully make a few points. Mathematical realism is not identified with a Platonism of archetypal objects outside spacetime. There are abstract things, such as the meaning of a word or a student's desire to graduate that are not outside spacetime. On the other hand, the issue of the independent existence of mathematical objects such as numbers can be interpreted as independence from human consciousness or "minds" or as independence from language. In the nominalist and fictionalist tradition, originating in Bentham's writings, it is possible to interpret these objects as semantic referents of discourse itself. Thus, a mathematical theory may describe structural features of reality that have been discovered, it may describe features of human consciousness constitutive of experience, somewhat in a Kantian sense, or it may describe fictional objects of discourse itself that, under an instrumentalist perspective, function to manipulate reality "as if" they existed in a literal sense.

I was under the assumption that most practicing mathematicians are Platonists, if only, perhaps, closeted Platonists, as when pressed to intellectually justify their position, they might try to articulate a Structuralist justification, but would still 'feel' and 'act' as Platonists in their everyday practice.

Nice video, I don't get how there is equality in th enumbers, 1=1implies 1(on the left,having this property) = 1(on the right having this property) which seems incorrect to me, in nature everything is unique and the space and time are two properties that two or more objects can not share. In the abstraction, when we say two red apples, I can imagine two left and right apples , one up and down apples, or apple means not the exactly same physical object in the minds of people, even in the mind of thesame person it may change the by time and place.There seems no one to one correspondence in numbers. Therefore mathematics has a flaw to explain and model the universe.

are there any prerequisites for this course?

Philosophy of mathematics

Should have 3M views

32:00 cursed auto-captioning "so daddy can identify"

A number is the measure of a magnitude.

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@1:03:30 How about the definition of a number is the intersection of all possible definitions of a number 😂

Also: you didn't talk about Intuitionism/Constructivism. Is it because they are a very minority view?

I am from MSc Mathematics. I am interested in resurch in algebraic topology and jyametry number theory and set theory. Please guide me. I bill from India.

Here is a Wittgensteinian take: Q: How many fingers do you have on your left hand? A: Five. Numbers are ANSWERS TO A CERTAIN CLASS OF QUESTIONS.

number three is the class of all three elements sets? Oh man!!! Where is our lovely mathematics going? Saying number three is the class of all three elements sets is like: question: Who are you? Answer: I'm the one who it is me.

haha dude looks like walter levin......

numbers are the building blocks of the universe, they are tangible things that make up the fabric of space and time, the universe is a flowing sea of these things, like a giant granular computer. well, if the word 'computer' even means anything other than an blob of these tangible 'things', hence why we can create computers ourselves, in which we can create and manipulate tangible worlds using numbers. numbers. numbers? numbers. NUMBERS! numbers :( Numbers :) numbers are, prove me that they arent.

The only numbers are the Rational Numbers. There are no other numbers.

An interesting overview of numbers. But awful, terrible teaching technique. Rapid fire speech (at 0.75 speech it is bearable), with the dullest of delivery. I feel sorry for the young students paying sky-high fees to sit through this lecture, and then being forced to buy his text.

And this is why philosophy is meaningless. It's people of limited intellect talking about things that they don't understand and can't do themselves.