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I am reading "Probability and Stochastics" by Cinlar. Here is the following quote in the preface of the book.

As Martin Barlow put it once, mathematics attracts us because the need to memorize is minimal.

Without focusing too much on the affective-emotive aspect — attracts — but rather on the cognitive aspect of requiring minimal memorization I wonder:
What is the meaning of this quote? Could we infer that mathematics is some internal facility of mind, as intuitionism suggests? What diminished role does memory has to play then?

Rushi
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Kartik Pandey
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    Maybe the author refers to the platitude that it is enough to remember the axioms of a theory and everything else can be derived from them by deduction... – Mauro ALLEGRANZA Jan 22 '24 at 14:41
  • @MauroALLEGRANZA then again not all deductions are worth mentioning or remembering. and then again not all deductions naturally occur as effortlessly as the theory fits in the head, after consuming it. Mathematics does become like second nature unlike other subject fields. – Kartik Pandey Jan 22 '24 at 14:49
  • @SonOfThought Do you wish to contest that math requires to memorize a lot? Even omitting the "alot", could you contest this proposition that unlike other subjects mathematics relies the least on empiricism. Of course, I am not Martin Barlow himself. So why are you trying to explore it? why not? – Kartik Pandey Jan 22 '24 at 15:43
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    This is not a question about philosophy; it's a question about English. It's perfectly ordinary in English for the grammatical subject of a sentence to be an inanimate object, or even a concept; and this is not a claim that the subject has agency. Casual speech in English simply doesn't distinguish "to attract" from "to be attractive". (Aside: I grow weary of finding philosophy.se questions on the sidebar that turn out to be really questions about English. The site moderators and curators really should take this more seriously.) – Karl Knechtel Jan 23 '24 at 02:15
  • Ive made a small edit to address the close-voters and commenters that are fixating on the attracts aspect. Needless to say you are free to revert/modify my addition as you see fit. – Rushi Jan 23 '24 at 07:25

5 Answers5

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I think what it's referring to is this idea that it's quite beautiful, somehow, that all of mathematics arises with just a very few premises. You don't need to memorize because nothing is arbitrary, everything naturally pops out once you accept just the bare minimum axioms.

Compare that to something like, say, specializing in European history. In European history, no amount of axioms of history can help you derive who was the monarch of Luxembourg in 1722. You either know the fact, or you do not.

There's something beautiful about the fact that in Mathematics, all truths are in principle accessible and derivable.

TKoL
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    Your first sentence, while commonly believed in certain groups, is provably false. This is one of the main consequences of Godel's Incompleteness Theorems. Even without formal proof, simply noting that there are multiple different kinds of geometry using different sets of axioms quickly shows that you need a fairly large set of axioms to merely encompass all of geometry. – TimothyAWiseman Jan 23 '24 at 00:53
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    For reference: what the incompleteness theorems tell us is that there are unprovable sentences, but what this answer relies on is the (correct) statement that many statements can be proved from a small number of axioms. So Godel is irrelevant. Though one might want to study many axiomatic systems (rarely I suspect - most geometers would work in some fairly simple framework) there are still fewer axioms than results by quite a margin. – Francis Davey Jan 23 '24 at 06:26
  • The core of this answer is correct, and the fact that one can derive a lot from a small number of axioms is true and beautiful. The claim that all of mathematics arises with just a very few premises is provably false even for large values of "very few". – TimothyAWiseman Jan 23 '24 at 17:28
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Guess it just refers to the experience of school children or learners. Who's "work" with regards to "learning" is usually comprised of submitting something to memory by constant repetition, which is a long and tedious process that often isn't much fun. While math, is in so far different as it's actually more effective to be lazy.

The less memorizing that you can get away with and do the job, the better. Where in the most extreme you only actually memorize a few axioms and algorithms and everything else can be deduced from that.

haxor789
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  • Personally, I still practiced a lot of questions to prepare for math tests. Even through higher education, students are advised to maximally "practice" the back exercises of the books, previous papers.

    Math does require repetition but it is deliberate deduction in contexts (questions) than anything.

     – Kartik Pandey Jan 22 '24 at 15:49
  • I have a belief that - specifically for math. We attempt many deductions (questions) that we intrinsically develop an "understanding" of the subject which fosters "intuition". Of course, if I am developing all theorems from deduction that innately, then what would mathematicians do? We have a grand body of knowledge in mathematics, which is intrinsically not available to us. But after the consumption of the theory through study, it becomes so ingrained that we could fluently manage it. Like a learnt language. unlike other subjects which require hard facts. – Kartik Pandey Jan 22 '24 at 15:59
  • "intrinsically not available to us" is backed by the fact that notions like "continuity" and "probability" are not immediately ideated by mind. but when they do, I need to not remember them as proofs or theorem. They just occur.

    like a semantic of a word - it just pops in right situations. love. hate. peace.

     – Kartik Pandey Jan 22 '24 at 16:08
  • I think I will satisfy myself my concluding that mathematics functions like a language. I have not recorded the entire dictionary to say words I wish to. – Kartik Pandey Jan 22 '24 at 16:19
  • Well sure you're supposed to practice and remember stuff because you're supposed to perform these operations in time and without necessarily thinking about them, as you are likely doing those on a job and are not supposed to spend a large amount of time reinventing the wheel but is that quintessential to math? – haxor789 Jan 23 '24 at 07:50
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    Also at least in theory math is a lot more interconnected. So while in lots of domains you have sort of an island knowledge where information is linked to a specific context but doesn't transfer well to other contexts, at least in theory math is one giant construct. – haxor789 Jan 23 '24 at 12:19
  • Also what do you mean by intrinsically? Like at birth? Without being told? Like you would empirically develop an understanding of continuity and probability, right? – haxor789 Jan 23 '24 at 12:19
  • Thanks for the reply. Yes, I meant that. Personally, I faced a lot of trouble understanding concepts like probability and continuity. Even after reading the theorems and solving problems. But once I did, I was able to "see" it at many places. Even outside of mathematics. It was like "grasping" a semantic. relevant – Kartik Pandey Jan 23 '24 at 16:38
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Something like the ZFC axioms, with much else forgotten, would provide a wealth of knowledge. But is this not also true of the concept of Turing Machines, which are conceptually even simpler and quicker to state. And in physics, take the case of amplituhedrons, "processes that would have required more Feynman diagrams than atoms in the universe [can] now be represented by a single diagram". They help describe scattering amplitudes, "the bread and butter of quantum field theory"^. And quantum field theory is at leading edge of our most accurate scientific theories ever, allowing us to predict to down 17 decimal places.

Perhaps this is a stubborn response to the author's point, but Newtonian Mechanics attracts me because of the lack of memorization needed as well. Math can't be removed from physics, but the physical relations between velocity and acceleration don't require much memorization. And for History, there's less memorization than commonly jested. But, I may equivocating on their use of memorization, or maybe not. It feels like a nice quip, but not true to further analysis.

I’d further add Plato’s Problem, the idea that language is not learned but unlocked, might make the authors point even harder to maintain. We don’t memorize as much as we think we do. We have some kind of capacity for at least language, which undergirds much of our knowledge. In sum, to narrowly pick out mathematics seems questionable.

^ https://www.ams.org/journals/notices/201802/rnoti-p167.pdf

J Kusin
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Unlike other subjects mathematics relies the least on empiricism

Yes This! At least...

In Theory

  • In physics g is 9.8 m/s2
  • In history WWII was declared by Britain on Sept 3 1939
  • Doctors need to know about the ins and outs of zillions of medicines

etc., — all arbitrary, memory-demanding facts.

It may seem that physics is more analytical than the others. So ok one can push back g to G = 6.674×10−11 N⋅m2/kg2 which is at least as arbitrary as g.
Curiously, when I looked up G (which I of course dont know by heart) the lead starts by saying its an empirical physical constant

So yes, one could argue that there are no such arbitrary-s in math.
At least in theory...

In Practice

Things are quite different in practice

Definitions

Things like groups, fields, rings have certain standard definitions. They could have been defined with some permutation, eg the definition we give to rings could have been given to groups, etc

Terminology

Imaginary numbers are an old bugbear of mine.
I suspect that real numbers cause bigger problems in the sense that mathematicians tend to treat members of ℝ as real (in the ordinary sense), forgetting that the fights between Cantor and Kronecker going all the way to Hilbert and the warring schools of math in the twentieth century start with this.

Abuse of notation

Mathematicians regularly create notations and then for some convenience reason or other abuse them. eg see Abuse of the asymptotic big O

All these choices are empirical-arbitrary and need to be remembered as such by practitioners

So...

In Summary

Like all human activities, mathematics has a conceptual — aka Platonic — core. I guess its reasonable to argue that this is more so for math than most other fields.

Also like all human activities its done by humans! viz. bumbling, fumbling foibled people like us. Once those foibles get ossified into a standard the standard needs to be learnt as «tradition».

Perhaps one could say:

  1. Math is platonic in-the-small and empiric in-the-large. The definitions and shape that Euler gave to graph theory could and would be different if someone else had created the theory in a different time and clime. But the specific proof of the Königsberg bridge problem would be the same modulo the new definitions
  2. [To the specific question]
    Platonic ≆ analytic and is derivable
    Empiric ≆ synthetic ie arbitrary and demands memory
Rushi
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  • I concur. Although, as is evident, I have a strong bias for Intuitionism. Platonism seems out-right ridiculous and there is no reason to not believe all Empiricism could be "serendipity", yet it keeps justification of beliefs, formidable. Intuitionism provides that mutual ground that keeps scientists sane. However as a young man can't afford strong biases. I proclaim a Skeptic stance. This, as I foresee, would make a very grumpy and disgruntled old man. Not something to look forward to. – Kartik Pandey Jan 23 '24 at 06:47
  • Nice to meet you again – Kartik Pandey Jan 23 '24 at 06:47
  • @KartikPandey All these math-ism fights that date from late 19th century did not exist for the first two millennia of math when much of the mosyt significant math was developed. The trouble with Platonism is that the name was stolen by the post-cantoreans. I am reasonably persuaded that Plato would not have approved of the wanton set theory that Cantor started and what followed. Nor Gauss. et al. In fact one could also assert that CS arose from attempting to set right the apple cart the Cantor overturned – Rushi Jan 23 '24 at 06:52
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    Which is to say that the divisions between Platonists and intuitionists was non existent or nominal until the post-Cantorists started doing theology and calling it math. [Gordon on the axiom of choice: This sir, is not math its theology ]. And obscuring this by calling themselves Platonists. Since the name has stuck the best we can do is distinguish platonist from Platonist. See my longer answer – Rushi Jan 23 '24 at 06:56
  • @KartikPandey Personally if I had a choice, I'd call the (modern) platonist a locutionist. ie just because the Russel-set expression — R = {x | x ∉ x} — parses correctly it doesnt mean it denotes anything. Or maybe a Chomskyist after Chomsky's famous example of a sentence that parses grammatically but means nothing: Colorless green ideas sleep furiously – Rushi Jan 23 '24 at 07:15
  • I read your long answer. I pondered upon it. Specifically, for philosophy, I have decided to maintain a very pragmatic outlook. I "subjectively" consume philosophy to consolidate a belief system (similar to jungian introverted thinking ). And thus, I have chosen to reject Platonism, without denying it in the absolute. A platonic realm could be, but to accept it as belief sends the locus of control outside of me. rejecting it or (by denial of the excluded middle) shunning it, preserves my sanity. – Kartik Pandey Jan 23 '24 at 08:16
  • The above comment was in defence of me calling Platonism "ridiculous". Moreover, of course, I have confused notions of platonism with Platonism. Besides, I have developed crazy math theories in my earlier life. To say less, I used to believe Math came from a third godly realm, I used to search clues in the Purans and Shashtra. I convinced myself that infinity is the symbolic incarnate of Shiv . Conclusively, I exhausted myself and discontinued it all. – Kartik Pandey Jan 23 '24 at 08:31
  • A bit nit picky but the gravitational constant on earth this small "g" depends on where you are on the surface and WWII officially started with Hitler's invasion of Poland Britain and France declared war by ultimatum on the 3rd of September and it took until 1940 for Churchill to succeed Neville Chamberlain. – haxor789 Jan 23 '24 at 09:31
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    @haxor789 Thanks for the correction re WWII. Ive incorporated. As for g yes Im aware its a "varying constant"!! – Rushi Jan 23 '24 at 09:49
  • @KartikPandey I guess the difficulty arises because people take these ideas too literally. ie if you ask where this Platonic world can be found no good answer is available when actually the conceptual nub of platonism is that its truths are non spatial and timeless (see abstract) so the question of "where" is moot – Rushi Jan 23 '24 at 09:53
  • @Rushi A few gifted can entertain philosophy as it is. However, I am interested in your beliefs. They are peculiar and yet they stand. To you, what is god? Is only mathematics platonic? or is it these "feelings" / qualia also exists and relevates from a different realm, one which is absolute? – Kartik Pandey Jan 23 '24 at 09:58
  • @KartikPandey Id recommend opening a chat if you like a freer discussion. But to answer your question pithily (for here): I am not an atheist; I am not a theist. The former follows from the axiom that I did not make myself; the latter from the axiom that any conception I may have of ultimate reality is guaranteed to be wildly wrong. Of course this is a rough sketch. In advaita : drishti-shristi vada, I created myself this morning when on awakening I created an individual - me, a world - an imagination of a collection of "everything", and God - when I imagined something beyond. But start chat – Rushi Jan 23 '24 at 10:04
  • @Rushi yes, please. – Kartik Pandey Jan 23 '24 at 10:47
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Academic mathematics does not require a lot of memorization. If you understand the underlying principles, you can generally re-derive anything that you need to know for academic or research based mathematics. There is a certain amount of memorization that is useful, but it is far, far less than in many other fields.

This is in contrast to many other academic fields where memorization is essential. As my wife (a teacher of history) would be quick to point out, the further you go into history the less memorization is necessary and the more analysis is expected. Nonetheless, a historian must still know a huge number of key facts about their subject area to be able to work effectively.

Notably, there is no finite set of axioms you can memorize that would then, even in principle, allow you to re-derive the remainder of mathematics. This is one of the key implications of Godel's incompleteness theorem. The number of axioms needed is infinite. With that said, certain specific fields can by definition be completely described by a relatively simple set of axioms, such as Tarski's Axioms for Geometry.

TimothyAWiseman
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