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How has mathematics gotten so abstract?

139 points by thadt 1y ago

thadt 1y ago
This reminds of of that one time when I was on a date with a girl from the history department who somehow bemusedly sat through my entire mini-lecture on comparing infinite sets. Twenty years and three kids later, she'll still occasionally look me straight in the eye and declare "my infinity is bigger than your infinity."
- dcchuck 1y ago
  This is the type of romcom I'd watch ;)
  - reverendjames 1y ago
    You want to watch him and his wife?
- muragekibicho 1y ago
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  - deleted 1y ago
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  - halfmatthalfcat 1y ago
    There’s something called “not giving a fuck” that works in those situations. The crux of it though is you need to “know thyself” or you’ll be forever your worst critic and enemy.
    - HPsquared 1y ago
      Also you're being open and readable to the other person. You're not being deceptive or putting on a show, which is usually what rustles people's jimmies.
  - KeplerBoy 1y ago
    You would not. People love hearing about the things you care about as long as you can present them in interesting ways. Try it!
  - Cthulhu_ 1y ago
    I wouldn't go on Hinge if that's the default experience.
    - fknorangesite 1y ago
      It is not, of course.
  - MontyCarloHall 1y ago
    Get this incel trolling out of here. No, all that would happen if your date weren’t interested in math/whatever you’re into is a polite message “I had a great time but I don’t think we have much in common” and leave it at that.
    - DonHopkins 1y ago
      His oddly specific fear of being "plastered on a bunch of facebook new-york-dating-experience groups" sounds like women have had to warn each other about him before, and it probably wasn't about his interest in math, but something much worse.
- grues-dinner 1y ago
  Fittingly this is roughly the same vintage as your relationship then: https://youtu.be/BipvGD-LCjU https://youtu.be/BipvGD-LCjU
- deleted 1y ago
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- wvlia5 1y ago
  Once I taught the binomial coefficient formula to a girl after sex
  - huflungdung 1y ago
    [dead]
  - mbork_pl 1y ago
    The Fibonacci sequence might have been more appropriate.
    - lo_zamoyski 1y ago
      Not if they were using contraception.
  - grues-dinner 1y ago
    "So you see if the chance of pregnancy is constant per..uh..encounter, and given that the condom just broke, we're on a spectrum from the chance of a second round roughly doubling the odds but the overall chance is still small, or it doesn't make much difference anyway. Either way, the numbers say we should go again."
  - dominicrose 1y ago
    that's not too abstract, I can see how this formula applies to sex I tried using if for this: https://adventofcode.com/2023/day/12 https://adventofcode.com/2023/day/12 but computer said no
- btilly 1y ago
  I'm curious. Did either of you ever notice the implicit philosophical assumptions that you have to make to come to the conclusion that one infinity can be larger than another? Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.
  - dragonwriter 1y ago
    > Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way. That’s not at all true at the level where you are dealing with different infinities, usually, which tends to come after the (usually, fairly early) part dealing with proofs and the fact that all mathematics is dealing with “unprovable intellectual leaps” which are encoded into axioms, and everything in math which is provable is only provable based on a particular chosen set of axioms. It may be true that math beyond that basic level doesn’t make a point of going back and explicitly reviewing that point, but it is just kind of implicit in everything later.
    - btilly 1y ago
      I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism. Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.
      - godelski 1y ago
        > I guarantee that a naive presentation doesn't actually include the axioms But you said "modern math courses". Are you now talking about a casual conversation? I mean the OP's story is that his wife just liked listening to him talk about his passions. > Uncountable need not mean more. Sure. But that doesn't mean that there aren't differing categories. However you slice it, we can operate on these things in different ways. Real or not the logic isn't consistent between these things but they do fall out into differing categories. If you're trying to find mistakes in the logic does it not make sense to push it at its bounds? Look at the Banach-Tarski Paradox. Sure, normal people hear about it and go "oh wow, cool." But when it was presented in my math course it was used as a discussion of why we might want to question the Axiom of Choice, but that removing it creates new concerns. Really the "paradox" was explored to push the bounds of the axiom of choice in the first place. They asked "can this axiom be abused?" And the answer is yes. Now the question is "does this matter, since infinity is non-physical? Or does it despite infinity being non-physics?" You seem to think mathematicians, physicists, and scientists in general believe infinities are physical. As one of those people, I'm not sure why you think that. We don't. I mean math is a language. A language used because it is pedantic and precise. Much the same way we use programming languages. I'm not so sure why you're upset that people are trying to push the bounds of the language and find out what works and doesn't work. Or are you upset that non-professionals misunderstand the nuances of a field? Well... that's a whole other conversation, isn't it...
- deleted 1y ago
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- gnulinux 1y ago
  Wow, I did a very similar thing on the first date with my now wife. I explained the halting problem, and Godel's incompleteness theorems. We also talked about her (biomedical) research, so it wasn't a one sided conversation. I think dominating on a first date is a risk (which I was mindful of) but just being yourself, and talking about something you're truly passionate about is the key.
- charlieyu1 1y ago
  I taught my wife simplex algorithm for linear programming and she forgot all of it Turns out I’m neither good in maths nor teaching
- pfdietz 1y ago
  Way back then, calculus was a culture war battleground. Bishop Berkeley famously argued the foundations of calculus weren't any better that those of theology. This sort of thing motivated much work into shoring them up, getting rid of infinitesimals and the like (or, later, making infinitesimals rigorous in nonstandard analysis). https://en.wikipedia.org/wiki/The_Analyst https://en.wikipedia.org/wiki/The_Analyst
- Waterluvian 1y ago
  This is the sweetest thing ever and I hope you feel those butterflies even now sharing this story.
bmitc 1y ago
What else is it supposed to do?
- wwweston 1y ago
  I think this is a really good question, and the answer might be that ideally you move up and down the ladder of abstraction, learning from concrete examples in some domains, then abstracting across them, then learning from applying the abstractions, then abstracting across abstractions, then cycling through the process.
TZubiri 1y ago
[flagged]
- jumpingscript 1y ago
  https://en.wikipedia.org/wiki/Micha%C5%82_Zalewski https://en.wikipedia.org/wiki/Micha%C5%82_Zalewski
- ak1ng 1y ago
  Could be him? https://en.wikipedia.org/wiki/Micha%C5%82_Zalewski https://en.wikipedia.org/wiki/Micha%C5%82_Zalewski
- roel_v 1y ago
  Did you bother to google his handle? While I don't know his pure mathematics credentials, he's nerd-famous enough to not warrant an introduction. In fact, you not recognizing it says something about you.
  - jlongr 1y ago
    >he's nerd-famous enough to not warrant an introduction What is nerd-famous supposed to be. He's at the center of some subjective in-group that exists in your head?
    - wizzwizz4 1y ago
      He wrote the American Fuzzy Lop fuzzer, which was extremely influential – pretty much put fuzzing on the map.
    - TZubiri 1y ago
      To be fair, we are on hacker news. I did once use on of his programs, American Fuzzy Lopper (fake advertisement lawsuit incoming if its not american). So he is not nobody apparently
- fridek 1y ago
  I'm curious how you managed to find nothing on lcamtuf. He's one of the most famous Polish hackers from the 90s, then one the best security researchers Google had. Even if you live under a rock, the substack has an "about" section. If it wasn't for Michał I'd probably be a farmer today.
doe88 1y ago
My mental representation of this phenomenon is like inverted Russian dolls: you start by learning the inner layers, the basics, and as you mature, you work your way into more abstractions, more unified theories, more structures, adding layers as you learn more and more. Adding difficulty but this extreme refinement is also very beautiful. When studying mathematics I like to think of all these steps, all the people, and centuries of trial and errors, refinements it took to arrive where we are now.
hodgehog11 1y ago
I feel like a great deal more credit should be given to Cauchy and his school, but I understand the tale is long enough. The Peano axioms are pretty nifty though. To get a better appreciation of the difficulty of formally constructing the integers as we know them, I recommend trying the Numbers Game in Lean found here: https://adam.math.hhu.de/ https://adam.math.hhu.de/
Tazerenix 1y ago
>Today, mathematics is regarded as an abstract science. Pure mathematics is regarded as an abstract science, which it is by definition. Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences. >On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs. Mathematicians use intuition routinely at all levels of investigation. This is captured for example by Tao's famous stages of rigour (https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...). Mathematicians require that their intuition is useful for mathematics: if intuition disagrees with rigour, the intuition must be discarded or modified so that it becomes a sharper, more useful razor. If intuition leads one to believe and pursue false mathematical statements, then it isn't (mathematical) intuition after all. Most beginners in mathematics do not have the knowledge to discern the difference (because mathematics is very subtle) and many experts lack the patience required to help navigate beginners through building (and appreciating the importance of) that intuition. The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way. The mainstream view in mathematics is that infinite sets, especially ones as pedestrian as the naturals or the reals, are not particularly weird after all. Once one develops the aforementioned mathematical intuition (that is, once one discards the naive, human-centric notion that our intuition about finite things should be the "correct" lens through which to understand infinite things, and instead allows our rigorous understanding of infinite sets to inform our intuition for what to expect) the confusion fades away like a mirage. That process occurs for all abstract parts of mathematics as one comes to appreciate them (expect, possibly, for things like spectral sequences).
- pdpi 1y ago
  > Pure mathematics is regarded as an abstract science, which it is by definition. I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.
rob74 1y ago
How has mathematics gotten so abstract? My understanding was that mathematics was abstract from the very beginning. Sure, you can say that two cows plus two more cows makes four cows, but that already is an abstraction - someone who has no knowledge of math might object that one cow is rarely exactly the same as another cow, so just assigning the value "1" to any cow you see is an oversimplification. Of course, simple examples such as this can be translated into intuitive concepts more easily, but they are still abstract.
- TuringTest 1y ago
  > My understanding was that mathematics was abstract from the very beginning. It wasn't; but that's a common misunderstanding from hundreds of centuries of common practice. So, how has maths gotten so abstract? Easy, it has been taken over by abstraction astronauts(1), which have existed throghout all eras (and not just for software engineering). Mathematics was created by unofficial engineers as a way to better accomplish useful activities (guessing the best time of year to start migrating, and later harvesting; counting what portion of harvest should be collected to fill the granaries for the whole winter; building temples for the Pharaoh that wouldn't collapse...) But then, it was adopted by thinkers that enjoyed the activity by itself and started exploring it by sheer joy; math stopped representing "something that needed doing in an efficient way", and was considered "something to think about to the last consecuences". Then it was merged into philosophy, with considerations about perfect regular solids, or things like the (misunderstood) metaphor of shadows in Plato's cave (which people interpreted as being about duality of the essences, when it was merely an allegory on clarity of thinking and explanation). Going from an intuitive physical reality such as natural numbers ("we have two cows", or "two fingers") to the current understanding of numbers as an abstract entity ("the universe has the essence of number 'two' floating beyond the orbit of Uranus"(2)) was a consequence of that historical process, when layers upon layers of abstraction took thinkers further and further away from the practical origins of math. [1] https://www.joelonsoftware.com/2001/04/21/dont-let-architecture-astronauts-scare-you/ https://www.joelonsoftware.com/2001/04/21/dont-let-architect... [2] https://en.wikipedia.org/wiki/Hyperuranion https://en.wikipedia.org/wiki/Hyperuranion
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jjgreen 1y ago
The number 1 is what a cow, a fox, a stone ... have in common, oneness. Mathematics is abstraction, written down.
- prmph 1y ago
  That's not obvious. - they are material objects - they are concepts I understand - they are sequences of letters - they are English words - ... Not sure why oneness is privileged as what they have in common, and their oneness is meaningless by itself. Oneness is a property that is only meaningful in relation to other concepts of objects.
  - jjgreen 1y ago
    A rock is not physically a material object, it is a region of space where the electrons, protons and neutrons are differently arranged, and that region is fuzzy, difficult to determine; but as physical beings, as monkeys, we recognise its oneness, that's necessary for our survival in this physical world, we see this blurred outline of a rock, we feel it's weight in our hand, we observe its practical difference from two rocks. Just as we recognise twoness in a pair of rocks, fish, apples, threeness in a triple of parrots, of carrots, we abstract those out into 1, 2, 3, ...
intrasight 1y ago
There was a time, not that long ago in human history, that zero was "so abstract".
- dist-epoch 1y ago
  It was a religious offense to talk about zero. https://cambriamathtutors.com/zero-christianity/ https://cambriamathtutors.com/zero-christianity/
- stonemetal12 1y ago
  Sure even 500 years ago negative numbers were "absurd" in western mathematics and even in eastern mathematics where they were used they were more thought of as credits and debts than just abstract numbers.
fidotron 1y ago
Unlike Zeno's famous example the paradox which does better at explaining the problem is https://en.wikipedia.org/wiki/Coastline_paradox https://en.wikipedia.org/wiki/Coastline_paradox which Mandelbrot seemed particularly keen on. The tendency towards excessive abstraction is the same as the use of jargon in other fields: it just serves to gatekeep everything. The history of mathematics (and science) is actually full of amateurs, priests and bored aristocrats that happened to help make progress, often in their spare time.
- azan_ 1y ago
  Theirs no such thing as excessive abstraction in math, because abstraction is the point. Is category theory “excessive abstraction” in your opinion?
  - fidotron 1y ago
    > because abstraction is the point. Formal reasoning is the point, which is not by itself abstraction. Someone else in this discussion is saying Euclid's Elements is abstract, which is near complete nonsense. If that is abstract our perception of everything except for the fundamental [whatever] we are formed of is an abstraction.
    - empath75 1y ago
      > Formal reasoning is the point, which is not by itself abstraction. What do you think "formal" means in that sentence. It means "formal" from the word "form". It is reasoning through pure manipulation of symbols, with no relation to the external world required.
      - fidotron 1y ago
        I love how you lot just redefine words to suit your purpose: https://www.etymonline.com/word/formal https://www.etymonline.com/word/formal "late 14c., "pertaining to form or arrangement;" also, in philosophy and theology, "pertaining to the form or essence of a thing," from Old French formal, formel "formal, constituent" (13c.) and directly from Latin formalis, from forma "a form, figure, shape" (see form (n.)). From early 15c. as "in due or proper form, according to recognized form," As a noun, c. 1600 (plural) "things that are formal;" as a short way to say formal dance, recorded by 1906 among U.S. college students." There's not a much better description of what Euclid was doing.
elAhmo 1y ago
Isn't this true for many other fields of study? Given the collective time put into it, easier stuff was already solved thousands of years ago, and people are not really left with something trivial to work on. Hence focusing on more and more abstract things as those are the only things left to do something novel.
- dist-epoch 1y ago
  You are right, the low hanging fruits were picked a long time ago. But also wrong, the easier stuff was solved INCORRECTLY thousands of years ago. But it takes advanced math to understand what was incorrect about it.
- currymj 1y ago
  two interesting cases: convex analysis and linear algebra are both relatively easy, concrete areas of mathematics. also beautiful and unbelievably useful. yet they didn't develop until the 19th century and didn't mature until the 20th.
iamwil 1y ago
It's always been abstract. They'll say to me, "Give me a concrete example with numbers!" I get what they're saying in practice. But numbers are abstract. They only seem concrete because you'd internalized the abstract concept.
falcor84 1y ago
I found it a bit ironic that the author introduced C code there as an aid, but didn't incorporate it into their argument. As I see it, code is exactly the bridge between abstract math and the empirical world - the process of writing code to implement your mathematical structure and then seeing if it gives you the output you expect (or better yet, with Lean, if it proves your proposition) essentially makes math a natural science again.
- Ar-Curunir 1y ago
  No, the correctness of your implementation is a mathematical statement about a computation running a particular computational environment, and can be reasoned about from first principles without ever invoking a computer. Whether your computation gives reasonable outputs on certain inputs says nothing (in general) about the original mathematics.
  - falcor84 1y ago
    While mathematics "can" be reasoned about from first principles, the history of math is chock-full of examples of professional mathematicians convinced by unsound and wrong arguments. I prefer the clarity of performing math experiments and validating proofs on a computer.
    - Ar-Curunir 1y ago
      Yes, but a C or Python program that “implements” a proof and which you test by running it on a few inputs is very different from a program in a interactive theorem prover like Rocq or Lean. In the latter, validity is essentially decided by type-checking, not execution
aristofun 1y ago
How has blog posts authors gotten so uneducated or/and clickbaiting? Math in its core has always been abstract. It’s the whole point.
- The_suffocated 1y ago
  > Math in its core has always been abstract. It’s the whole point. I don't think so. E.g. there may be some abstractions in numerical linear algebra, but the subject matter has always been quite concrete.
  - aristofun 1y ago
    It is not a matter of what you think it is a logical fact, part of the definition if you will. What you call concrete - were the origins of math as we know it. Geometry, astronomy, metaphysics etc they all had in common fundamental abstract thing that we call math today. Saying “math got abstract” is like saying “a tree got wooden”. Because when it was a seed - it wasn’t yet a tree in a full sense.
deleted 1y ago
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s20n 1y ago
I believe mathematics was much tamer before Georg Cantor's work. If I had to pick a specific point in history when maths got "so abstract", it would be the introduction of axiomatic set theory by Zermelo. I personally cannot wrap my head around Cantor's infinitary ideas, but I'm sure it makes perfect sense to people with better mathematical intuition than me.
boxerab 1y ago
The French Bourbaki school certainly had a large influence on increasing abstraction in math, with their rallying cry "Down With Triangles". The more fundamental reason is that generalizing a problem works; it distills the essence and allows machinery from other branches of math to help solve it. "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies." -- Stefan Banach
nivter 1y ago
I believe that abstraction is recursive in nature which creates multiple layers of abstract ideas leading to new areas or insights. For instance our understanding of continuity and limit led to calculus, which when tied to the (abstract) idea of linearity led to the idea of linear operator which explains various phenomena in the real world surprisingly well.
- masklinn 1y ago
  You could say that abstraction is a step or a ladder: by climbing on an abstraction you can see new goals and opportunities, possibly out of reach until you build yet new steps.
johngossman 1y ago
I think the title is a little tongue in cheek. The rest of the blog post develops the Foundations of arithmetic in a clear, well-grounded manner. This is probably a really good introduction for someone about to take a Foundations course. I say this having just Potter's "Set Theory and it's Philosophy" which covers the same material (and a lot more obviously) in 300 some pages. Another good introduction is Frederic Schuller's YouTube lectures, though already there you can start to see the over abstraction.
pgustafs 1y ago
The definition of bijection is much more interesting than comparing cardinals. Many everyday use cases where (structure-preserving) bijections make it clear that two apriori different objects can be treated similarly. More generally, mathematics is experimental not just in the sense that it can be used to make physical predictions, but also (probably more importantly) in that definitions are "experiments" whose outcome is judged by their usefulness.
daxfohl 1y ago
One could also say the opposite. It's not abstract at all, just a set of rules and their implications. Plausibly the least abstract thing there is. On the other hand, two cookies plus three cookies, what even is a cookie? What if they're different sizes? Do sandwich cookies count as one or two? If you cut one in half, does you count it as two cookies now? All very abstract. Just give me some concrete definitions and rules and I'll give you a concrete answer.
The_suffocated 1y ago
Discussions of this sort can easily get chaotic, because people tend to conflate intuitiveness and concreteness. Sometimes the whole point of abstraction is to make a concept clearer and more intuitive. The distinction between polynomial function and polynomial is an example.
btilly 1y ago
Proposed rule: People writing about the history of mathematics, should learn something about the history of mathematics. Mathematicians didn't just randomly decide to go to abstraction and the foundations of mathematics. They were forced there by a series of crises where the mathematics that they knew fell apart. For example Joseph Fourier came up with a way to add up a bunch of well-behaved functions - sin and cos - and came up to something that wasn't considered a function - a square wave. The focus on abstraction and axiomatization came after decades of trying to repair mathematics over and over again. Trying to retell the story in terms of the resulting mathematical flow of the ideas, completely mangles the actual flow of events.
- crabbone 1y ago
  Yeah... The article doesn't even attempt to answer the question in its title. It's just a watered down Intro to Mathematics 101.
- coffeeaddict1 1y ago
  I have to disagree with this. Modern (pure) mathematics is abstract and very often completely detached from practical applications because of culture and artistic inspiration. There is no "objectivity" driving modern pure mathematics. It exists mostly because people like thinking about it. Any connection to the real world is often a coincidence or someone outside the field noticing that something (really just a tiny-tiny amount) in pure maths could be useful. > forced there by a series of crises where the mathematics that they knew fell apart This can be said to be true of those working in foundations, but the vast majority of mathematicians are completely uninterested in that! In fact, most mathematicians today probably can't cite you the set-theoretic (or any other foundation) axioms that they use every day, if you ask them point-blank.
jmount 1y ago
None of that was even the abstract stuff. It is all models of sizes, order, and inclusion (integers, cardinals, ordinals, sets). Not the nastier abstractions of partial orders, associativity, composition and so on (lattices, categories, ...).
- lambdasquirrel 1y ago
  And yet it all circles back. We used Peano arithmetic when doing C++ template metaprogramming anytime a for loop from 0..n was needed. It was fun and games as long as you didn't make a mistake because the compiler errors would be gnarly. The Haskell people still do stuff like this, and I wouldn't be surprised if someone were doing it in Scala's type system as well. Also, the PLT people are using lattices and categories to formalize their work.
yuppiemephisto 1y ago
I like Peano, but he was using Grassmann's definition of natural numbers
tphyahoo2 1y ago
Just drop the axiom of infinity and quit whining. https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism
- drdeca 1y ago
  Can one do QFT in an ultrafinitistic foundations? My guess is no. Also, I don’t think ZF sans the axiom of infinity works as an ultrafinitistic theory? It still has every natural number, just not the set of all of them.
ogogmad 1y ago
"Indeed, persistently trying to relate the foundations of math to reality has become the calling card of online cranks." <-- Hm??? I'm getting self-conscious. Details?
susam 1y ago
This article explores a particular kind of abstractness in mathematics, especially the construction of numbers and the cardinalities of infinite sets. It is all very interesting indeed. However, the kind of abstractness I most enjoy in mathematics is found in algebraic structures such as groups and rings, or even simpler structures like magmas and monoids. These structures avoid relying on specific types of numbers or elements, and instead focus on the relationships and operations themselves. For me, this reveals an even deeper beauty, i.e., different domains of mathematics, or even problems in computer science, can be unified under the same algebraic framework. Consider, for example, the fact that the set of real numbers forms a vector space over the set of rationals. Can it get more abstract than that? We know such a vector space must have a basis, but what would that basis even look like? The existence of such a basis (Hamel basis) is guaranteed by the axioms and proofs, yet it defies explicit description. That, to me, is the most intriguing kind of abstractness! Despite being so abstract, the same algebraic structures find concrete applications in computing, for example, in the form of coding theory. Concepts such as polynomial rings and cosets of subspaces over finite fields play an important role in error-correcting codes, without which modern data transmission and storage would not exist in their current form.
- griffzhowl 1y ago
  When I was learning me a Haskell I had a great time when I realised that as long as my type was a monoid I could freely chain the operations together purely because of associativity
trinsic2 1y ago
>Next, consider the time needed for Achilles to reach the yellow dot; once again, by the time he gets there, the turtle will have moved forward a tiny bit. This process can be continued indefinitely; the gap keeps getting smaller but never goes to zero, so we must conclude that Achilles can’t possibly win the race. Am i daft, eventually (Very soon) Achilles would over take the turtles position regardless of how far it moved... I am missing something?
- m_dupont 1y ago
  you're not, the proof is a famous error known as zenos paradox. Its only an apparent paradox, and indeed it's been disproven by observing that things do in fact move
  - trinsic2 1y ago
    Wow this is some serious over complication. How can anyone mix Philosophy and Mathematics? They are not even in the same ball park.. Even with infinity. Its just something that cant be understood in the mind, IMHO.
  - griffzhowl 1y ago
    I like the humourous way of putting it, but of course Zeno and his contemporaries knew that things moved - that's exactly why this seemed to be a paradox. Seemingly secure reasoning results in a conclusion that's obviously false. To resolve the paradox, you have to show what's wrong with the reasoning, not just observe the obviously false conclusion.
lottin 1y ago
I wish the scroll bar was a little less invisible.
BrandoElFollito 1y ago
I used to be a physicist and I love math for the toolbox it provides (mostly Analysis). It allows to solve a physical model and make predictions. When I was studying, I always got top marks in Analysis. Then came Algebra, Topology and similar nightmares. Oh crap, that was difficult. Not really because of the complexity, but rather because of abstraction, an abstraction I could not take to physics (I was not a very good physicist either). This is the moment I realized that I will never be "good in maths" and that will remain a toolbox to me. Fast forward 30 years, my son has differentials in high school (France, math was one of his "majors"). He comes to me to ask what the fuck it is (we have a unhealthy fascination for maths in France, and teach them the same was as in 1950). It is only when we went from physical models to differentials that it became clear. We did again the trip Newton did - physics rocks :)
initramfs 1y ago
This article can also be written as "The unreasonable effectiveness of abstraction in mathematics."
Animats 1y ago
Infinity is a convenience that pays off in terseness. There's constructive mathematics, but it's wordy and has lots of cases. You can escape undecidablity if you give up infinity. Most mathematicians consider that a bad trade.
wduquette 1y ago
I once walked around Westwood with my future wife telling her all about Karp-reductions of NP-complete problems. Somehow we now have four kids.
fooker 1y ago
My hypothesis for this is the disconnect between mathematics and fields like physics and theoretical computer science. We likely need new mathematics for making progress in physics or ..say.. have a better understanding of the PvsNP kind of problems, but very few high caliber mathematicians are motivated to do this. Which makes sense, as it’s way easier and prestigious to define and solve your own abstract problems, publish one paper per grad student per year and coast through research life.

139 points by thadt 1y ago

This reminds of of that one time when I was on a date with a girl from the history department who somehow bemusedly sat through my entire mini-lecture on comparing infinite sets. Twenty years and three kids later, she'll still occasionally look me straight in the eye and declare "my infinity is bigger than your infinity."

dcchuck 1y ago
This is the type of romcom I'd watch ;)
- reverendjames 1y ago
  You want to watch him and his wife?
muragekibicho 1y ago
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- deleted 1y ago
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- halfmatthalfcat 1y ago
  There’s something called “not giving a fuck” that works in those situations. The crux of it though is you need to “know thyself” or you’ll be forever your worst critic and enemy.
  - HPsquared 1y ago
    Also you're being open and readable to the other person. You're not being deceptive or putting on a show, which is usually what rustles people's jimmies.
- KeplerBoy 1y ago
  You would not. People love hearing about the things you care about as long as you can present them in interesting ways. Try it!
- Cthulhu_ 1y ago
  I wouldn't go on Hinge if that's the default experience.
  - fknorangesite 1y ago
    It is not, of course.
- MontyCarloHall 1y ago
  Get this incel trolling out of here. No, all that would happen if your date weren’t interested in math/whatever you’re into is a polite message “I had a great time but I don’t think we have much in common” and leave it at that.
  - DonHopkins 1y ago
    His oddly specific fear of being "plastered on a bunch of facebook new-york-dating-experience groups" sounds like women have had to warn each other about him before, and it probably wasn't about his interest in math, but something much worse.
grues-dinner 1y ago
Fittingly this is roughly the same vintage as your relationship then: https://youtu.be/BipvGD-LCjU https://youtu.be/BipvGD-LCjU
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wvlia5 1y ago
Once I taught the binomial coefficient formula to a girl after sex
- huflungdung 1y ago
  [dead]
- mbork_pl 1y ago
  The Fibonacci sequence might have been more appropriate.
  - lo_zamoyski 1y ago
    Not if they were using contraception.
- grues-dinner 1y ago
  "So you see if the chance of pregnancy is constant per..uh..encounter, and given that the condom just broke, we're on a spectrum from the chance of a second round roughly doubling the odds but the overall chance is still small, or it doesn't make much difference anyway. Either way, the numbers say we should go again."
- dominicrose 1y ago
  that's not too abstract, I can see how this formula applies to sex I tried using if for this: https://adventofcode.com/2023/day/12 https://adventofcode.com/2023/day/12 but computer said no
btilly 1y ago
I'm curious. Did either of you ever notice the implicit philosophical assumptions that you have to make to come to the conclusion that one infinity can be larger than another? Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way.
- dragonwriter 1y ago
  > Despite the fact that this was actively debated for decades, modern math courses seldom acknowledge the fact that they are making unprovable intellectual leaps along the way. That’s not at all true at the level where you are dealing with different infinities, usually, which tends to come after the (usually, fairly early) part dealing with proofs and the fact that all mathematics is dealing with “unprovable intellectual leaps” which are encoded into axioms, and everything in math which is provable is only provable based on a particular chosen set of axioms. It may be true that math beyond that basic level doesn’t make a point of going back and explicitly reviewing that point, but it is just kind of implicit in everything later.
  - btilly 1y ago
    I guarantee that a naive presentation doesn't actually include the axioms, and doesn't address the philosophical questions dividing formalism from constructivism. Uncountable need not mean more. It can mean that there are things that you can't figure out whether to count, because they are undecidable.
    - godelski 1y ago
      > I guarantee that a naive presentation doesn't actually include the axioms But you said "modern math courses". Are you now talking about a casual conversation? I mean the OP's story is that his wife just liked listening to him talk about his passions. > Uncountable need not mean more. Sure. But that doesn't mean that there aren't differing categories. However you slice it, we can operate on these things in different ways. Real or not the logic isn't consistent between these things but they do fall out into differing categories. If you're trying to find mistakes in the logic does it not make sense to push it at its bounds? Look at the Banach-Tarski Paradox. Sure, normal people hear about it and go "oh wow, cool." But when it was presented in my math course it was used as a discussion of why we might want to question the Axiom of Choice, but that removing it creates new concerns. Really the "paradox" was explored to push the bounds of the axiom of choice in the first place. They asked "can this axiom be abused?" And the answer is yes. Now the question is "does this matter, since infinity is non-physical? Or does it despite infinity being non-physics?" You seem to think mathematicians, physicists, and scientists in general believe infinities are physical. As one of those people, I'm not sure why you think that. We don't. I mean math is a language. A language used because it is pedantic and precise. Much the same way we use programming languages. I'm not so sure why you're upset that people are trying to push the bounds of the language and find out what works and doesn't work. Or are you upset that non-professionals misunderstand the nuances of a field? Well... that's a whole other conversation, isn't it...
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gnulinux 1y ago
Wow, I did a very similar thing on the first date with my now wife. I explained the halting problem, and Godel's incompleteness theorems. We also talked about her (biomedical) research, so it wasn't a one sided conversation. I think dominating on a first date is a risk (which I was mindful of) but just being yourself, and talking about something you're truly passionate about is the key.
charlieyu1 1y ago
I taught my wife simplex algorithm for linear programming and she forgot all of it Turns out I’m neither good in maths nor teaching
pfdietz 1y ago
Way back then, calculus was a culture war battleground. Bishop Berkeley famously argued the foundations of calculus weren't any better that those of theology. This sort of thing motivated much work into shoring them up, getting rid of infinitesimals and the like (or, later, making infinitesimals rigorous in nonstandard analysis). https://en.wikipedia.org/wiki/The_Analyst https://en.wikipedia.org/wiki/The_Analyst
Waterluvian 1y ago
This is the sweetest thing ever and I hope you feel those butterflies even now sharing this story.

bmitc 1y ago

What else is it supposed to do?

wwweston 1y ago
I think this is a really good question, and the answer might be that ideally you move up and down the ladder of abstraction, learning from concrete examples in some domains, then abstracting across them, then learning from applying the abstractions, then abstracting across abstractions, then cycling through the process.

TZubiri 1y ago

[flagged]

jumpingscript 1y ago
https://en.wikipedia.org/wiki/Micha%C5%82_Zalewski https://en.wikipedia.org/wiki/Micha%C5%82_Zalewski
ak1ng 1y ago
Could be him? https://en.wikipedia.org/wiki/Micha%C5%82_Zalewski https://en.wikipedia.org/wiki/Micha%C5%82_Zalewski
roel_v 1y ago
Did you bother to google his handle? While I don't know his pure mathematics credentials, he's nerd-famous enough to not warrant an introduction. In fact, you not recognizing it says something about you.
- jlongr 1y ago
  >he's nerd-famous enough to not warrant an introduction What is nerd-famous supposed to be. He's at the center of some subjective in-group that exists in your head?
  - wizzwizz4 1y ago
    He wrote the American Fuzzy Lop fuzzer, which was extremely influential – pretty much put fuzzing on the map.
  - TZubiri 1y ago
    To be fair, we are on hacker news. I did once use on of his programs, American Fuzzy Lopper (fake advertisement lawsuit incoming if its not american). So he is not nobody apparently
fridek 1y ago
I'm curious how you managed to find nothing on lcamtuf. He's one of the most famous Polish hackers from the 90s, then one the best security researchers Google had. Even if you live under a rock, the substack has an "about" section. If it wasn't for Michał I'd probably be a farmer today.

doe88 1y ago

My mental representation of this phenomenon is like inverted Russian dolls: you start by learning the inner layers, the basics, and as you mature, you work your way into more abstractions, more unified theories, more structures, adding layers as you learn more and more. Adding difficulty but this extreme refinement is also very beautiful. When studying mathematics I like to think of all these steps, all the people, and centuries of trial and errors, refinements it took to arrive where we are now.

hodgehog11 1y ago

I feel like a great deal more credit should be given to Cauchy and his school, but I understand the tale is long enough. The Peano axioms are pretty nifty though. To get a better appreciation of the difficulty of formally constructing the integers as we know them, I recommend trying the Numbers Game in Lean found here: https://adam.math.hhu.de/ https://adam.math.hhu.de/

Tazerenix 1y ago

>Today, mathematics is regarded as an abstract science. Pure mathematics is regarded as an abstract science, which it is by definition. Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences. >On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs. Mathematicians use intuition routinely at all levels of investigation. This is captured for example by Tao's famous stages of rigour (https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...). Mathematicians require that their intuition is useful for mathematics: if intuition disagrees with rigour, the intuition must be discarded or modified so that it becomes a sharper, more useful razor. If intuition leads one to believe and pursue false mathematical statements, then it isn't (mathematical) intuition after all. Most beginners in mathematics do not have the knowledge to discern the difference (because mathematics is very subtle) and many experts lack the patience required to help navigate beginners through building (and appreciating the importance of) that intuition. The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way. The mainstream view in mathematics is that infinite sets, especially ones as pedestrian as the naturals or the reals, are not particularly weird after all. Once one develops the aforementioned mathematical intuition (that is, once one discards the naive, human-centric notion that our intuition about finite things should be the "correct" lens through which to understand infinite things, and instead allows our rigorous understanding of infinite sets to inform our intuition for what to expect) the confusion fades away like a mirage. That process occurs for all abstract parts of mathematics as one comes to appreciate them (expect, possibly, for things like spectral sequences).

pdpi 1y ago
> Pure mathematics is regarded as an abstract science, which it is by definition. I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.

rob74 1y ago

How has mathematics gotten so abstract? My understanding was that mathematics was abstract from the very beginning. Sure, you can say that two cows plus two more cows makes four cows, but that already is an abstraction - someone who has no knowledge of math might object that one cow is rarely exactly the same as another cow, so just assigning the value "1" to any cow you see is an oversimplification. Of course, simple examples such as this can be translated into intuitive concepts more easily, but they are still abstract.

TuringTest 1y ago
> My understanding was that mathematics was abstract from the very beginning. It wasn't; but that's a common misunderstanding from hundreds of centuries of common practice. So, how has maths gotten so abstract? Easy, it has been taken over by abstraction astronauts(1), which have existed throghout all eras (and not just for software engineering). Mathematics was created by unofficial engineers as a way to better accomplish useful activities (guessing the best time of year to start migrating, and later harvesting; counting what portion of harvest should be collected to fill the granaries for the whole winter; building temples for the Pharaoh that wouldn't collapse...) But then, it was adopted by thinkers that enjoyed the activity by itself and started exploring it by sheer joy; math stopped representing "something that needed doing in an efficient way", and was considered "something to think about to the last consecuences". Then it was merged into philosophy, with considerations about perfect regular solids, or things like the (misunderstood) metaphor of shadows in Plato's cave (which people interpreted as being about duality of the essences, when it was merely an allegory on clarity of thinking and explanation). Going from an intuitive physical reality such as natural numbers ("we have two cows", or "two fingers") to the current understanding of numbers as an abstract entity ("the universe has the essence of number 'two' floating beyond the orbit of Uranus"(2)) was a consequence of that historical process, when layers upon layers of abstraction took thinkers further and further away from the practical origins of math. [1] https://www.joelonsoftware.com/2001/04/21/dont-let-architecture-astronauts-scare-you/ https://www.joelonsoftware.com/2001/04/21/dont-let-architect... [2] https://en.wikipedia.org/wiki/Hyperuranion https://en.wikipedia.org/wiki/Hyperuranion
- 1y ago

jjgreen 1y ago

The number 1 is what a cow, a fox, a stone ... have in common, oneness. Mathematics is abstraction, written down.

prmph 1y ago
That's not obvious. - they are material objects - they are concepts I understand - they are sequences of letters - they are English words - ... Not sure why oneness is privileged as what they have in common, and their oneness is meaningless by itself. Oneness is a property that is only meaningful in relation to other concepts of objects.
- jjgreen 1y ago
  A rock is not physically a material object, it is a region of space where the electrons, protons and neutrons are differently arranged, and that region is fuzzy, difficult to determine; but as physical beings, as monkeys, we recognise its oneness, that's necessary for our survival in this physical world, we see this blurred outline of a rock, we feel it's weight in our hand, we observe its practical difference from two rocks. Just as we recognise twoness in a pair of rocks, fish, apples, threeness in a triple of parrots, of carrots, we abstract those out into 1, 2, 3, ...

intrasight 1y ago

There was a time, not that long ago in human history, that zero was "so abstract".

dist-epoch 1y ago
It was a religious offense to talk about zero. https://cambriamathtutors.com/zero-christianity/ https://cambriamathtutors.com/zero-christianity/
stonemetal12 1y ago
Sure even 500 years ago negative numbers were "absurd" in western mathematics and even in eastern mathematics where they were used they were more thought of as credits and debts than just abstract numbers.

fidotron 1y ago

Unlike Zeno's famous example the paradox which does better at explaining the problem is https://en.wikipedia.org/wiki/Coastline_paradox https://en.wikipedia.org/wiki/Coastline_paradox which Mandelbrot seemed particularly keen on. The tendency towards excessive abstraction is the same as the use of jargon in other fields: it just serves to gatekeep everything. The history of mathematics (and science) is actually full of amateurs, priests and bored aristocrats that happened to help make progress, often in their spare time.

azan_ 1y ago
Theirs no such thing as excessive abstraction in math, because abstraction is the point. Is category theory “excessive abstraction” in your opinion?
- fidotron 1y ago
  > because abstraction is the point. Formal reasoning is the point, which is not by itself abstraction. Someone else in this discussion is saying Euclid's Elements is abstract, which is near complete nonsense. If that is abstract our perception of everything except for the fundamental [whatever] we are formed of is an abstraction.
  - empath75 1y ago
    > Formal reasoning is the point, which is not by itself abstraction. What do you think "formal" means in that sentence. It means "formal" from the word "form". It is reasoning through pure manipulation of symbols, with no relation to the external world required.
    - fidotron 1y ago
      I love how you lot just redefine words to suit your purpose: https://www.etymonline.com/word/formal https://www.etymonline.com/word/formal "late 14c., "pertaining to form or arrangement;" also, in philosophy and theology, "pertaining to the form or essence of a thing," from Old French formal, formel "formal, constituent" (13c.) and directly from Latin formalis, from forma "a form, figure, shape" (see form (n.)). From early 15c. as "in due or proper form, according to recognized form," As a noun, c. 1600 (plural) "things that are formal;" as a short way to say formal dance, recorded by 1906 among U.S. college students." There's not a much better description of what Euclid was doing.

elAhmo 1y ago

Isn't this true for many other fields of study? Given the collective time put into it, easier stuff was already solved thousands of years ago, and people are not really left with something trivial to work on. Hence focusing on more and more abstract things as those are the only things left to do something novel.

dist-epoch 1y ago
You are right, the low hanging fruits were picked a long time ago. But also wrong, the easier stuff was solved INCORRECTLY thousands of years ago. But it takes advanced math to understand what was incorrect about it.
currymj 1y ago
two interesting cases: convex analysis and linear algebra are both relatively easy, concrete areas of mathematics. also beautiful and unbelievably useful. yet they didn't develop until the 19th century and didn't mature until the 20th.

iamwil 1y ago

It's always been abstract. They'll say to me, "Give me a concrete example with numbers!" I get what they're saying in practice. But numbers are abstract. They only seem concrete because you'd internalized the abstract concept.

falcor84 1y ago

I found it a bit ironic that the author introduced C code there as an aid, but didn't incorporate it into their argument. As I see it, code is exactly the bridge between abstract math and the empirical world - the process of writing code to implement your mathematical structure and then seeing if it gives you the output you expect (or better yet, with Lean, if it proves your proposition) essentially makes math a natural science again.

Ar-Curunir 1y ago
No, the correctness of your implementation is a mathematical statement about a computation running a particular computational environment, and can be reasoned about from first principles without ever invoking a computer. Whether your computation gives reasonable outputs on certain inputs says nothing (in general) about the original mathematics.
- falcor84 1y ago
  While mathematics "can" be reasoned about from first principles, the history of math is chock-full of examples of professional mathematicians convinced by unsound and wrong arguments. I prefer the clarity of performing math experiments and validating proofs on a computer.
  - Ar-Curunir 1y ago
    Yes, but a C or Python program that “implements” a proof and which you test by running it on a few inputs is very different from a program in a interactive theorem prover like Rocq or Lean. In the latter, validity is essentially decided by type-checking, not execution

aristofun 1y ago

How has blog posts authors gotten so uneducated or/and clickbaiting? Math in its core has always been abstract. It’s the whole point.

The_suffocated 1y ago
> Math in its core has always been abstract. It’s the whole point. I don't think so. E.g. there may be some abstractions in numerical linear algebra, but the subject matter has always been quite concrete.
- aristofun 1y ago
  It is not a matter of what you think it is a logical fact, part of the definition if you will. What you call concrete - were the origins of math as we know it. Geometry, astronomy, metaphysics etc they all had in common fundamental abstract thing that we call math today. Saying “math got abstract” is like saying “a tree got wooden”. Because when it was a seed - it wasn’t yet a tree in a full sense.

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s20n 1y ago

I believe mathematics was much tamer before Georg Cantor's work. If I had to pick a specific point in history when maths got "so abstract", it would be the introduction of axiomatic set theory by Zermelo. I personally cannot wrap my head around Cantor's infinitary ideas, but I'm sure it makes perfect sense to people with better mathematical intuition than me.

boxerab 1y ago

The French Bourbaki school certainly had a large influence on increasing abstraction in math, with their rallying cry "Down With Triangles". The more fundamental reason is that generalizing a problem works; it distills the essence and allows machinery from other branches of math to help solve it. "A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies." -- Stefan Banach

nivter 1y ago

I believe that abstraction is recursive in nature which creates multiple layers of abstract ideas leading to new areas or insights. For instance our understanding of continuity and limit led to calculus, which when tied to the (abstract) idea of linearity led to the idea of linear operator which explains various phenomena in the real world surprisingly well.

masklinn 1y ago
You could say that abstraction is a step or a ladder: by climbing on an abstraction you can see new goals and opportunities, possibly out of reach until you build yet new steps.

johngossman 1y ago

I think the title is a little tongue in cheek. The rest of the blog post develops the Foundations of arithmetic in a clear, well-grounded manner. This is probably a really good introduction for someone about to take a Foundations course. I say this having just Potter's "Set Theory and it's Philosophy" which covers the same material (and a lot more obviously) in 300 some pages. Another good introduction is Frederic Schuller's YouTube lectures, though already there you can start to see the over abstraction.

pgustafs 1y ago

The definition of bijection is much more interesting than comparing cardinals. Many everyday use cases where (structure-preserving) bijections make it clear that two apriori different objects can be treated similarly. More generally, mathematics is experimental not just in the sense that it can be used to make physical predictions, but also (probably more importantly) in that definitions are "experiments" whose outcome is judged by their usefulness.

daxfohl 1y ago

One could also say the opposite. It's not abstract at all, just a set of rules and their implications. Plausibly the least abstract thing there is. On the other hand, two cookies plus three cookies, what even is a cookie? What if they're different sizes? Do sandwich cookies count as one or two? If you cut one in half, does you count it as two cookies now? All very abstract. Just give me some concrete definitions and rules and I'll give you a concrete answer.

The_suffocated 1y ago

Discussions of this sort can easily get chaotic, because people tend to conflate intuitiveness and concreteness. Sometimes the whole point of abstraction is to make a concept clearer and more intuitive. The distinction between polynomial function and polynomial is an example.

btilly 1y ago

Proposed rule: People writing about the history of mathematics, should learn something about the history of mathematics. Mathematicians didn't just randomly decide to go to abstraction and the foundations of mathematics. They were forced there by a series of crises where the mathematics that they knew fell apart. For example Joseph Fourier came up with a way to add up a bunch of well-behaved functions - sin and cos - and came up to something that wasn't considered a function - a square wave. The focus on abstraction and axiomatization came after decades of trying to repair mathematics over and over again. Trying to retell the story in terms of the resulting mathematical flow of the ideas, completely mangles the actual flow of events.

crabbone 1y ago
Yeah... The article doesn't even attempt to answer the question in its title. It's just a watered down Intro to Mathematics 101.
coffeeaddict1 1y ago
I have to disagree with this. Modern (pure) mathematics is abstract and very often completely detached from practical applications because of culture and artistic inspiration. There is no "objectivity" driving modern pure mathematics. It exists mostly because people like thinking about it. Any connection to the real world is often a coincidence or someone outside the field noticing that something (really just a tiny-tiny amount) in pure maths could be useful. > forced there by a series of crises where the mathematics that they knew fell apart This can be said to be true of those working in foundations, but the vast majority of mathematicians are completely uninterested in that! In fact, most mathematicians today probably can't cite you the set-theoretic (or any other foundation) axioms that they use every day, if you ask them point-blank.

jmount 1y ago

None of that was even the abstract stuff. It is all models of sizes, order, and inclusion (integers, cardinals, ordinals, sets). Not the nastier abstractions of partial orders, associativity, composition and so on (lattices, categories, ...).

lambdasquirrel 1y ago
And yet it all circles back. We used Peano arithmetic when doing C++ template metaprogramming anytime a for loop from 0..n was needed. It was fun and games as long as you didn't make a mistake because the compiler errors would be gnarly. The Haskell people still do stuff like this, and I wouldn't be surprised if someone were doing it in Scala's type system as well. Also, the PLT people are using lattices and categories to formalize their work.

yuppiemephisto 1y ago

I like Peano, but he was using Grassmann's definition of natural numbers

tphyahoo2 1y ago

Just drop the axiom of infinity and quit whining. https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism

drdeca 1y ago
Can one do QFT in an ultrafinitistic foundations? My guess is no. Also, I don’t think ZF sans the axiom of infinity works as an ultrafinitistic theory? It still has every natural number, just not the set of all of them.

ogogmad 1y ago

"Indeed, persistently trying to relate the foundations of math to reality has become the calling card of online cranks." <-- Hm??? I'm getting self-conscious. Details?

susam 1y ago

This article explores a particular kind of abstractness in mathematics, especially the construction of numbers and the cardinalities of infinite sets. It is all very interesting indeed. However, the kind of abstractness I most enjoy in mathematics is found in algebraic structures such as groups and rings, or even simpler structures like magmas and monoids. These structures avoid relying on specific types of numbers or elements, and instead focus on the relationships and operations themselves. For me, this reveals an even deeper beauty, i.e., different domains of mathematics, or even problems in computer science, can be unified under the same algebraic framework. Consider, for example, the fact that the set of real numbers forms a vector space over the set of rationals. Can it get more abstract than that? We know such a vector space must have a basis, but what would that basis even look like? The existence of such a basis (Hamel basis) is guaranteed by the axioms and proofs, yet it defies explicit description. That, to me, is the most intriguing kind of abstractness! Despite being so abstract, the same algebraic structures find concrete applications in computing, for example, in the form of coding theory. Concepts such as polynomial rings and cosets of subspaces over finite fields play an important role in error-correcting codes, without which modern data transmission and storage would not exist in their current form.

griffzhowl 1y ago
When I was learning me a Haskell I had a great time when I realised that as long as my type was a monoid I could freely chain the operations together purely because of associativity

trinsic2 1y ago

>Next, consider the time needed for Achilles to reach the yellow dot; once again, by the time he gets there, the turtle will have moved forward a tiny bit. This process can be continued indefinitely; the gap keeps getting smaller but never goes to zero, so we must conclude that Achilles can’t possibly win the race. Am i daft, eventually (Very soon) Achilles would over take the turtles position regardless of how far it moved... I am missing something?

m_dupont 1y ago
you're not, the proof is a famous error known as zenos paradox. Its only an apparent paradox, and indeed it's been disproven by observing that things do in fact move
- trinsic2 1y ago
  Wow this is some serious over complication. How can anyone mix Philosophy and Mathematics? They are not even in the same ball park.. Even with infinity. Its just something that cant be understood in the mind, IMHO.
- griffzhowl 1y ago
  I like the humourous way of putting it, but of course Zeno and his contemporaries knew that things moved - that's exactly why this seemed to be a paradox. Seemingly secure reasoning results in a conclusion that's obviously false. To resolve the paradox, you have to show what's wrong with the reasoning, not just observe the obviously false conclusion.

lottin 1y ago

I wish the scroll bar was a little less invisible.

BrandoElFollito 1y ago

I used to be a physicist and I love math for the toolbox it provides (mostly Analysis). It allows to solve a physical model and make predictions. When I was studying, I always got top marks in Analysis. Then came Algebra, Topology and similar nightmares. Oh crap, that was difficult. Not really because of the complexity, but rather because of abstraction, an abstraction I could not take to physics (I was not a very good physicist either). This is the moment I realized that I will never be "good in maths" and that will remain a toolbox to me. Fast forward 30 years, my son has differentials in high school (France, math was one of his "majors"). He comes to me to ask what the fuck it is (we have a unhealthy fascination for maths in France, and teach them the same was as in 1950). It is only when we went from physical models to differentials that it became clear. We did again the trip Newton did - physics rocks :)

initramfs 1y ago

This article can also be written as "The unreasonable effectiveness of abstraction in mathematics."

Animats 1y ago

Infinity is a convenience that pays off in terseness. There's constructive mathematics, but it's wordy and has lots of cases. You can escape undecidablity if you give up infinity. Most mathematicians consider that a bad trade.

wduquette 1y ago

I once walked around Westwood with my future wife telling her all about Karp-reductions of NP-complete problems. Somehow we now have four kids.

fooker 1y ago

My hypothesis for this is the disconnect between mathematics and fields like physics and theoretical computer science. We likely need new mathematics for making progress in physics or ..say.. have a better understanding of the PvsNP kind of problems, but very few high caliber mathematicians are motivated to do this. Which makes sense, as it’s way easier and prestigious to define and solve your own abstract problems, publish one paper per grad student per year and coast through research life.