What is a number?

Joel David Hamkins, Professor of Logic, Oxford University
This lecture is based on chapter 1 of my book, Lectures on the Philosophy of Mathematics, published with MIT Press, mitpress.mit.e....
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.
See lecture course information, including the schedule of topics, at jdh.hamkins.org....

[@DrThalesAlexandre]

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

[@joeldavidhamkins5484]

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.

[@mrmetaphysics9457]

it is not common sense to consider zero as a natural number!

[@rhetoric5173]

Professor, have you thought about how arithmetic savants fit into your thoughts about numbers?

[@one_logic]

​@@mrmetaphysics9457When dealing with axiomatic mathematics, 0 is commonly thought of as a natural number

[@Israel2.3.2]

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

[@YM-cw8so]

Lectures in university just gives a broad sense of what that field is doing, what's special about university is their seminars and the resources including library and professors, in other words, doing your own research is pretty better than listening to these lectures

[@FidelMontalvoburela]

0:46

[@FidelMontalvoburela]

Mostrar pantallRe 1:27 sultado de la búsqueda

[@FidelMontalvoburela]

🎉🎉 😂🎉 3:33

[@danascully1248]

Profs gotta get paid. If everything is free for everyone, some ppl just won't go to school

[@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." .

[@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.

[@romanbesel4759]

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!

[@berkeleycodingacademy7015]

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.)

[@cjfthistle]

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

[@namelastname2449]

True

[@exantefacto4245]

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.

[@nothingposted9056]

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

[@primefactor5417]

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

[@john94139]

mathematical logic, too!!!

[@radientbeing]

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

[@lunavicta4172]

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.

[@ritvicpaarekh6963]

That could be due to him knowing what a prime number is and that the roundness acts as a sense to determine what is prime.

[@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.

[@MariaVarona-xh9zg]

Outstanding

[@DigitalGnosis]

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)?

[@joeldavidhamkins5484]

I've just seen this comment now. I address dialethism and paraconsistency, but only very briefly, in the lecture What is Proof? I'm sorry about the camera orientation---I really need a proper video producer. Perhaps I'll start with a proper tripod.

[@АндрейДенькевич]

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

[@roberthvistendahl8635]

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.

[@leodu561]

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?

[@manueldelrio7147]

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.

[@joeldavidhamkins5484]

I used to think this also, except that I have come to realize that there are far more formalists than I had realized. My current view is that there is quite a wide spectrum of philosophical views held by mathematicians regarding mathematical ontology. But also, to my way of thinking, the question of structuralism can be considered orthogonal to platonism, in the sense that one can be structuralist whether or not one is platonist and vice versa. Platonism has to do with the ontological question of the real existence of mathematical objects, and structuralism has to do with the focus of mathematical attention and whether it is to be placed on individual objects or instances or whether it is to be treated as invariant under isomorphism.

[@jonahellison4229]

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.

[@schmetterling4477]

You should probably wait with that question until the category revolution has redefined mathematics in more rational terms than the ones that were derived from set theory. From my naive understanding finite cardinals are boring like integers (plus zero) and infinite cardinals are not numbers in an ordinary sense but more like equivalence classes. Not sure why one would want to make one like the other.

[@kunalsingh48]

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

[@tommyphillips1030]

In base 16, you represent the first 16 numbers as 0,1,2,..,9,a,b,..f. (You need 16 distinct symbols to do this). Note that 15 is now represented by "f" and so is a palindrome.

[@sonarbangla8711]

The foundations of numbers were never defined as Wildberger, Cauchy maintain. For a long time mathematicians couldn't define 'infinity'. Until Cantor managed some kind of a two dimensional array of numbers, that is insufficient to philosophers. The cardinals were able to identify various types of infinities, but infinity remained undefined to most. Mathematical logic stemmed out of Cantor, Russel, Hilbert, Godel etc based on recursive functions and relations hiding behind undefined Metamathematics. What resulted is unproven undecidability of mathematics. What mathematics recently proved with the help of Reimann Zeta functions are a string of correspondences between mathematics and physical reality, based on complex numbers. Complex number i is defined as a ratio between effect and cause, proved by Tristan Needham in VISUAL COMPLEX ANALYSIS.

[@CriticalDispatches]

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

[@columbus8myhw]

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"?

[@columbus8myhw]

The same for striving for coordinate-free geometric proofs

[@thinkGrey_]

thank you love from india ...

[@wojciechaleksanderwooszyn2920]

Great!

[@yarochkindmitry7631]

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=)

[@Edkahmed]

are there any prerequisites for this course?

[@joeldavidhamkins5484]

Not really, no. The lectures were aimed at Oxford philosophy students preparing for the Phil Maths exam paper.

[@Edkahmed]

@@joeldavidhamkins5484 Thank You! it means a lot to me that one of my fav philosophers alive responded to me.

[@namelastname2449]

Should have 3M views

[@JayDee-b5u]

A number is the measure of a magnitude.

[@Blinkwing]

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

[@manueldelrio7147]

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

[@joeldavidhamkins5484]

The lecture series will be eight lectures, and that topic will arise in the lecture on proof.

[@pinkraven4402]

3

[@ZiroOne-hw7iw]

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.

[@EsatBargan]

Allen Sarah Miller Sandra Taylor Mary

[@abhishektrivedi8296]

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.

[@hilaleristi5513]

Özet Joel David Hamkins’in Oxford’daki matematik felsefesi dersleri, sayıların doğasını ve farklı sistemlerde nasıl temsil edilebileceğini inceliyor. Konular 📚 Hamkins’in dersleri, öğrenciler için geniş bir perspektif sunuyor. 🔢 Sayıların farklı sayı sistemlerinde nasıl temsil edildiği tartışılıyor. 📜 Platonizm, Mantıksallık ve Yapısalcılık gibi felsefi bakış açıları irdeleniyor. 🌌 Cantor-Hume ilkesi ve Dedekind aksiyomları ele alınıyor. 🎲 Sayılar, oyunlar olarak kavramsallaştırılıyor. 🔄 Yapısalcılık, matematiksel nesnelerin sistem içindeki yapısal rolünü vurguluyor. 🧠 Psikoloji ve bilişsel bilimlerin sayı kavramını anlama üzerindeki etkileri tartışılıyor. Açıklamalar 🧩 Felsefi Perspektifler: Platonizm, mantıksallık ve yapısalcılık gibi farklı felsefi akımlar, sayıların doğasını çeşitli şekillerde yorumluyor. Bu, matematiksel düşüncenin derinleşmesine katkı sağlıyor. 🔍 Çeşitli Tanımlar: Sayıların ne olduğu konusunda farklı görüşler ve tanımlar var. Bu çeşitlilik, matematiğin dinamik yapısını gösteriyor ve yeni tartışmalara yol açıyor. ⚖ Cantor-Hume İlkesi: Bu ilke, sayıların varlığını ve doğasını sorgularken, matematik felsefesinde önemli bir temel oluşturuyor. 🧠 Bilişsel Bilimler: Psikoloji ve bilişsel bilimler, sayıların nasıl anlaşıldığını etkileyerek, matematiksel düşüncenin doğasına dair yeni perspektifler sunuyor. 🎲 Sayılar ve Oyunlar: Sayıların oyunlar olarak düşünülmesi, matematiksel yapının eğlenceli ve etkileşimli bir yönünü ortaya koyuyor. 🔄 Yapısalcılığın Önemi: Yapısalcılık, matematiksel nesnelerin bireysel kimliklerinden ziyade, sistem içindeki rollerine odaklanarak matematiksel anlayışı derinleştiriyor. ❓ Soruların Açık Ucu: Sonuç olarak, sayıların ne olduğu sorusu hala açık ve süregelen bir keşif ve tartışma alanı olarak kalıyor.

[@aakashchakraborty3673]

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

[@paschalcharles6097]

I think mathematicians do not know math at all

[@BehroozCompani-fk2sx]

Philosophy of mathematics??? Numbers are names we have invented for counting things. All math is just addition. Subtraction, multiplication, division are all abbreviation for addition. 3 x 3 = 3 +3+3 etc... Don't make a big deal of math. It scares people not to do math. 😂😂😂😂😂

[@turncloak.]

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.

[@YM-cw8so]

Can you show me a physical number?

[@NewCalculus]

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

[@SisypheanRoller]

What do you mean?

[@NewCalculus]

@@SisypheanRoller Search for "discovering the concept of number a personal journey" and then click on the Academia link to see what I mean.

[@schmetterling4477]

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.

[@NewCalculus]

Nonsense! Real philosophers as there were in Ancient Greece are the reason you have any mathematics at all today. Contemporary philosophers are fools - I'll grant you this. Philosophy is the king of all knowledge. It's possible you simply haven't learned any real philosophy?

[@god5535]

"Those who can, do. Those who can't teach." Found lecture to be the tedium and boredom of a generic textbook anyone can find in the library. Instead, if he had any interesting thing to say, instead of rehashing entire history of "numbers" without even addressing this relative term, I would have been fascinated. Maybe that's what University of Oxford/students want. And no wonder they end up brainwashed robots incapable of creativity or originality.

[@berkeleycodingacademy7015]

@@god5535 This professor is clearly a leader in his field given the reviews of his book, the reviews of these lectures, and his command over the material. Feynman was a gifted teacher, and one of the greatest physicists of all time. He taught brilliantly, and he won the Nobel Prize. As you know from your knowledge of mathematics, one counterexample is enough to disprove a general statement about teachers not being competent in their fields. With that said, I think the material is richer in subsequent lectures. Original insights from a professor are also not going to come in a first lecture. If you look the professor up online you will see that he has made original contributions in the field.

[@NewCalculus]

@@god5535 The guy presenting these videos is not a mathematician's rear end. He has no clue what it means to be a number and neither do any of his ignorant colleagues.

[@Dystisis]

And this (your comment) is why no one likes mathematicians.

[@erincarmody8562]

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.

[@randylayhee2427]

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.

[@swavekbu4959]

Everything returns to philosophy when thought about deeply enough.

[@belengaz3034]

Philosophy of mathematics

[@oldsachem]

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?

[@pmcate2]

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

[@varshneydevansh]

[@SeanAnthony-j7f]

❤❤❤❤❤

[@ibrahimkaya7684]

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.

[@sergioaramburu3965]

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.

[@MdMonsur-ec3yt]

🎉😊❤ 1:16 1:16

[@rhetoric5173]

Professor, have you considered arithmetic savants?

[@flux81]

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

[@oldsachem]

Fascinating overview

[@ryanfrizzell736]

Love it. love numbers.

[@zapazap]

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.

[@zapazap]

Of course the answer is in the form of a numeral or procedure -- but we can abstract from any particular response by considering a natural equivalence relationship amongst them.

[@zapazap]

I don't mean to assert that this satisfies all cases of 'what is a number?'. But if you want to teach mathematics while introducing the fewest ontological presumptions, this is a way to provide A coherent answer. After that, we can choose to try to develop our mathematics as far as we naturally can WITHOUT compelling the asking of problems which may be (in a Wittgensteinian sense) a problem that results FROM our (perhaps fruitless or misleading) choice to use language in a certain way.

[@YM-cw8so]

​@zapazap what is -1, square root of 2, then, are they numbers?

[@zapazap]

@@YM-cw8so the answer to the question "what is the length of the diagonal os a unit square? Is root 2

[@MichaelPaulCassianFlynn-jm7oy]

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.