Joel David Hamkins
@JDHamkins
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Mathematics and philosophy of the infinite. Professor of Logic @NotreDame @UniofOxford #ProofandtheArt #PhilMaths #InfinitelyMore #BookOfInfinity
University of Notre Dame
Joined November 2015
My book, Proof and the Art of Mathematics, MIT Press 2020, has been awarded the 2024 Daniel Solow Author's Award by the Mathematical Association of America. What a pleasure it has been to see the response to my book. #ProofAndTheArt.
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I once saw an incredible lecture in Berkeley by a historian of science, one of the best talks I have ever seen. He started his talk sitting on the desk amongst a huge pile of physics textbooks from history, each in its day "the best textbook of its time.".
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At the mathematics department of the University of Barcelona.
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Pick's theorem, a mathematical gem, a profound mathematical theorem, but with an elementary proof. I find it excellent for teaching how to write proofs, because one naturally approaches the general theorem in a series of easier cases. #ProofandtheArt
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What a punchline! And what an effective way to convey his main point. The implication was completely clear that of course we should expect that eventually this text also will come to be seen as fundamentally wrong on some issue or other.
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This lecture has stuck with me now for decades as an example of excellence in how to give a talk—I remember it clearly now decades later. Would anyone know who it likely was that I saw? This was Berkeley in the early 1990s, but I think he was likely an invited guest speaker.
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Finally, he got to the physics text used currently in Intro Physics classes. He explained how it had improved on its predecessor, and then said that he didn't know of any specific physics issue it got wrong. He silently placed it on top of the pile of earlier, flawed texts.
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I am preparing a poster and video about counting beyond infinity—how to to count to ω². Comments welcome—please criticize. Final version will appear later. #PhilMaths
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Mathematical induction must be legitimate, since it has always worked in the past.
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In my Infinity class, we covered the surreal numbers. I presented my image of the surreal tree. But I looked down sheepishly as it was displayed. I explained that I had spent many hours creating this image, but now felt something like the mother of an ugly child.
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The surreal numbers, ALL the numbers, great and small.
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And then the next. The speaker proceeded systematically through the historical procession of physics texts, highlighting how each had improved upon its predecessors, but also explaining in specific detail ways in which each of them was flawed.
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He began by showing us the text written by a student of Isaac Newton, which explained Newtonian forces and the motion of planets. Nevertheless, it had various mistakes and misunderstandings. He set it aside and moved to the next book, also excellent, but also flawed in some way.
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For the aspiring mathematician in your life, may I suggest a holiday gift:. Proof and the Art of Mathematics.Learn the art of mathematical proof-writing with a beautiful book, brimming with advice & inspiring mathematical results via interesting elementary proofs. #ProofandtheArt
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Some were flawed in their account of light, another had an etching of a comet, "drawn by a person who had *never* seen a comet." Eventually, the progression moved into the 20th century, but still the texts were wrong on various fundamental points, often now quite technical.
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It may have been a talk for the Berkeley Skeptics Society. I remember the speaker intoning of each text, that "it was the premier text of physics in its time.".
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How to use the research literature. When approaching a difficult mathematical problem, first think deeply upon it on your own, using all the ideas you can muster, before consulting the work of others. Push your own ideas as hard as you can first, and read only afterward.
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New book idea: .Ten proofs of Gödel incompleteness.Successive chapters will have all the various proofs I know, whether from computability via Turing, from fixed-point self-reference, from Tarski, from universal definitions, via Russell, via Berry's paradox, and so forth.
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Guess the theorem. #GuessTheTheorem .Post your answer before reading the thread. #InfinitelyMore
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Take my Philosophy of Mathematics final exam!.Post your answers in response to each question.
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My lectures begin tomorrow. Lectures on the Philosophy of Mathematics.Wednesdays 11-12 am (UK time) during Oxford Michaelmas Term. We meet on Zoom at: Lectures will be recorded and made public. #PhilMaths.
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Foreshadowing — the algebra of orders. Here is an addition table for some simple finite orders. Did you know that you can add and multiply any two orders? A+B means a copy of A with a copy of B above. Stay tuned for multiplication.
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Daughter's math homework tonight: expected number of coin flips to get n heads in a row. What an enjoyable conversation we had. What is your solution?.
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Truth and provability, a thread. Let us compare Tarski's theorem on the nondefinability of truth with Gödel's incompleteness theorem. Smullyan advanced the view that much of the fascination with Gödel's theorem should be better directed toward Tarski's theorem.
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This semester, I am teaching a new course on Infinity.
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The isomorphism philosophy: in any mathematical context, the genuinely mathematical ideas and properties are precisely those preserved by isomorphism.
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A Calculus 1a math problem my daughter had called me about to tell me her solution. Prove or refute: if a sequence of real numbers a_0, a_1, a_2, . has the property that for every number n>1, the subsequence a_n, a_2n, a_3n, . is convergent, then the sequence is convergent.
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1. Proof and the Art of Mathematics, for aspiring mathematicians to learn how to write proofs #ProofandtheArt. 2. Lectures on the Philosophy of Mathematics, philosophy grounded in mathematics.#PhilMaths
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The impossible badge. The Legendary badge is earned on MathOverflow by hitting the 200 reputation point limit on 150 separate days. Nobody has ever gotten this badge on MathOverflow, ever, and I have been trying for 14 years. Currently I have hit the limit on 145 occasions.
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Just informed by MIT Press that my book is in contract to be made available in Chinese translation.
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Three logicians go into a bar. The bartender asks, "Do you all want a beer?" .The first logician says, "I don't know." .The second logician says, "I don't know." .And the third logician says, "Yes.".
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I'll be giving lectures this term on the philosophy of mathematics. Lectures will be held on Zoom every Wednesday 11-12 am during term. I intend to record the lectures and make them available later. Topics will follow selections from my new book.
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My daughter had a nice problem in her high-school math club. Suppose you have 1000 white points and 1000 black points in the plane, no three collinear. Can you draw segments connecting them in pairs, from white to black, using each point just once, without any edges crossing?.
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Just heard that daughter is accepted at Caltech—yay! .She says she wants to be a mathematician, but there's still time, of course, for anything. Meanwhile, I think we had hints of her interests in younger days.
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The algebra of orders — now with multiplication! .Here is a multiplication table for some simple finite orders. Did you know that you can multiply any two orders? .AxB means B copies of A.
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Do you prefer whiteboards or blackboards when writing math?.
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My publisher aims to put Proof and the Art of Mathematics up for an MAA book award, given for impactful undergrad mathematics books. If you have used my book with students and would be willing to support the nomination, please contact me. #ProofAndTheArt.
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I made a thing. I am truly very sorry. it used to have many more eyes.
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OK, logic twits, we're holding the . Largest Tweetable Number contest!.Respond to this tweet with a tweet describing a specific natural number. Largest number wins. What have you got? .Prize: One million dollars*. *prize will be divided by the value of the winning number.
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I filled my book Proof and the Art of Mathematics with beautiful mathematical arguments and insightful figures. #ProofandtheArt.Make a gift to a mathematical loved one—any mathematically inclined person, of any age, can appreciate and learn from this book.
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Philosophy of Mathematics final exam—students did great! I am pleased. Post your answers below.
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Theorem. The real numbers are not interpretable in the complex numbers as a field. The result, well-known in model theory, often surprises mathematicians, who sometimes expect to easily define R in C. Yet, this isn't possible.
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Instructors, if you will be teaching an Introduction-to-Proofs or Transition-to-Advanced-Mathematics course next year, please consider using my book Proof and the Art of Mathematics as a text. #ProofandtheArt
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Kolmogorov complexity 🧵.The complexity K(x) of a number or string x is defined to be the size of the smallest program that is able to produce x as output. With Turing machines, we often measure the size of a program by the number of states.
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Do you consider these mathematical expressions meaningful? Nonsense? Totally fine? Meaningful, but bad style?
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Pick's theorem is a mathematical gem. One computes the area of a polygon in the integer lattice simply by counting vertices. The proof is elementary, but not obvious, and so I spent a chapter on it in my proof-writing book. #ProofandtheArt
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An Oxford University admissions interview question. You are a contestant on a game show, known for having perfectly logical contestants. There is another contestant, whom you've never met, but whom you can count on to be perfectly logical, just as logical as you are.
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My son, at the University of Bath, is putting together his engineering portfolio for various internship applications, and he included a photo of his greatest engineering accomplishment, from February.
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What are the real numbers, really?.An excerpt from my book, Lectures on the Philosophy of Mathematics. #InfinitelyMore #PhilMaths.
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Kafkaesque. Since leaving Oxford last year, I have been supervising my five Oxford D Phil students via Zoom. I planned to visit Oxford during Trinity term to meet with them and research colleagues. But I am now informed that meeting with my students would be considered "work" for.
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The problem with reading first before engaging on your own is that you may get trapped in another perspective, which could prevent you from realizing your key insight. But if you think first and then read, you combine your insight with prior knowledge, which may lead to progress.
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Stand up for your ignorance. In graduate school, I forced myself into the policy: admit ignorance freely when you don't fully understand something; ask questions, and ask again, until you do.
If you *know* you don't understand an idea that you encounter in graduate school, but people keep acting like it's obvious, and like you understand, DON'T PRETEND YOU UNDERSTAND! KEEP ASKING EVEN IF YOU FEEL STUPID. You don't want to end up like the adult version of this guy:
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As a Berkeley graduate student, I was TA for William Kahan teaching linear algebra, and will never forget his lecture in which he argued that matrix multiplication is defined the way it is precisely because of the chain rule.
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According to the naive theory of infinite cardinality:. 1. Different sizes of infinity? How absurd! All infinities are the same size. 2. The rationals are equinumerous with the natural numbers? How absurd! Obviously there are far more rational numbers.
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Cut a circular hole of radius 1 in a piece of paper. What is the largest size disc that you can pass through that hole? Allow yourself to fold the paper as much as you want, but not tear it.
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A puzzle posed by my wife Barbara: two people climb a staircase and then climb an escalator. One person rests a minute on the staircase and the other rests a minute on the escalator, but otherwise they climb stairs at the same rate. Who is faster or are they equally fast?.
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My PhD student, Bokai Yao @BokaiYao, has just passed his dissertation defense, defending the thesis, "Set theory with urelements", in which he establishes and separates a surprisingly rich hierarchy of urelement set theories, and develops the theory of forcing over urelement set
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The power set lattice on the natural numbers P(ℕ) under the subset relation. A work in progress.
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Confession: this is how I do all my proofs. 😗.
Paul Halmos' ingenious method for proving trigonometric identities
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Consider several structures on the natural numbers:. • Successor ⟨ℕ,0,S⟩. • Order ⟨ℕ,0,<⟩. • Addition ⟨ℕ,0,+⟩.What I claim is that each structure is definable in the next, but not conversely.
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It seems I shall be appearing on Lex Fridman's podcast.
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Physics question arising from my daughter's high-school physics class: .If you aim to break a window, should you rather throw a sticky lump of clay or a bouncy ball of the same mass? Or does it not matter?.
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Me: finishing seminar on Gödel incompleteness and Hilbert program. Student: Do you know the book Gödel, Escher, Bach?.Me: Oh yes, by Douglas Hofstadter. I read it when I was a kid. It was a big influence on me. Student: I didn't know it was that old!.Me: um. .
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Nice to be cited by Terry Tao (and Tamar Ziegler). Paper on the arxiv at
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A free extended excerpt from my book, Lectures on the Philosophy of Mathematics, an introduction for mathematicians and philosophers, grounded in mathematics. This week we consider Logicism, aiming to found mathematics in logic. #InfinitelyMore #PhilMaths.
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Next fall, I'll be teaching a graduate seminar on all aspects of the Gödel incompleteness phenomena. We'll follow my book-in-progress, Ten proof of Gödel incompleteness, with supplemental readings.
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A selection from the chapter on Relations and Functions in my book, Proof and the Art of Mathematics, helping aspiring mathematicians learn to write proofs. #ProofandtheArt Thanks for the great five-star reviews on Amazon!
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Can you solve my daughter's HS challenge math problem? There are 10 kids, assorted into various (possibly overlapping) groups. Each group has an odd number of people, but the intersection of any two has an even number. Show there are at most ten groups.
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Latest result: the Wordle number is independent of ZFC. It is consistent that it is strictly between aleph_0 and the the continuum. The wordle number w is the size of the smallest dictionary of infinite words (length ω, alphabet ω) that admits no finitely-winning strategy.
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The 16 binary truth functions. We know them all---we are not missing any.
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Mathematics instructors teaching Intro-to-Proofs next year—please consider my book, Proof and the Art of Mathematics. Beautiful mathematical theorems with elementary proofs. Diverse proof methods. Mathematical habits. Colorful figures. #ProofandtheArt.
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Chaos train quantifier logic. Can you match each formal statement with its corresponding phrase?
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My publisher intends to nominate my book for a certain math pedagogy award. Proof and the Art of Mathematics. If as a mathematics instructor you have used my book (in any capacity) with positive pedagogical results, could you kindly DM me? #ProofandtheArt
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In my book, I argue that it is a form of confirmation bias when considering the axiom of choice to look only at the controversial consequences of AC, without also considering the even more bizarre situations that AC rules out. #PhilMaths
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A great puzzle from my colleague Bill Child. A card is draw from an ordinary 52 card deck, and you aim to guess what it is. Before making your guess, you are allowed to ask one yes/no question about the card. Which of the 3 questions listed here will best improve your odds?.
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An imaginary mathematical history.🧵.I should like to sketch an imaginary mathematical history, an alternative history by which the continuum hypothesis (CH) might have come naturally to be seen as a core axiom of set theory and one furthermore necessary for ordinary mathematics.
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Advice for mathematicians. One of the mathematical habits I suggest in Proof and the Art of Mathematics #ProofandtheArt
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@maanow This is not mathematics. Mathematicians write their mathematical expressions in a clear manner, aiming to be understood, rather than relying on irritating or obscure notational rules. Add some parentheses to communicate clearly and say what you mean.
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I'll be giving graduate lectures/seminars this term on the philosophy of mathematics, joint with Timothy Williamson. Half the sessions will focus the philosophy of set theory. Details later. Tuesdays 2-4 TT19 in the Ryle Room. First reading is Maddy's book, Defending the Axioms.
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Illustrating the Banach-Tarski paradox by using rigid motion to produce two unit cubes from one! .
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Lectures on the Philosophy of Mathematics, MIT Press, is now available for pre-order. This is an introduction to the philosophy of mathematics, from a perspective grounded in mathematics, motivated by mathematical inquiry and practice. #PhilMaths.
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Today is book-release day---it's finally available!.Lectures on the Philosophy of Mathematics, an introduction to the philosophy of mathematics, grounded in mathematics. #PhilMaths.
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I once asked Ted Slaman, known for his excellent math talks, what is the secret? He replied, "You have to think like a comedian." He wasn't suggesting jokes, but rather, to plan the timing and the way ideas unfold, so as to enable your audience to reach a climax of insight.
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How the continuum hypothesis could have been a fundamental axiom.A simple historical thought experiment, showing how .we might have come to view CH as a fundamental axiom, one necessary for mathematics, indispensable even for calculus.
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