[HN Gopher] How has mathematics gotten so abstract?
___________________________________________________________________
How has mathematics gotten so abstract?
Author : thadt
Score : 111 points
Date : 2025-09-30 12:33 UTC (10 hours ago)
(HTM) web link (lcamtuf.substack.com)
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| thadt wrote:
| 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 wrote:
| This is the type of romcom I'd watch ;)
| grues-dinner wrote:
| Fittingly this is roughly the same vintage as your relationship
| then: https://youtu.be/BipvGD-LCjU
| wvlia5 wrote:
| Once I taught the binomial coefficient formula to a girl after
| sex
| mbork_pl wrote:
| The Fibonacci sequence might have been more appropriate.
| lo_zamoyski wrote:
| Not if they were using contraception.
| grues-dinner wrote:
| "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 wrote:
| 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 but computer said no
| btilly wrote:
| 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 wrote:
| > 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 wrote:
| 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 wrote:
| > 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...
| btilly wrote:
| Your guesses at what I seem to think are completely off
| base and insulting.
|
| When I say "modern math courses", I mean like the
| standard courses that most future mathematicians take on
| their way to various degrees. For all that we mumble ZFC,
| it is darned easy to get a PhD in mathematics without
| actually learning the axioms of ZFC. And without learning
| anything about the historical debates in the foundations
| of mathematics.
| zozbot234 wrote:
| The "philosophical questions" dividing formalism from
| constructivism are greatly overstated. The point of
| having those degrees of undecidability or uncountability
| is precisely to be able to say things like "even if you
| happen to be operating under strong additional
| assumptions that let you decide/count X, that still
| doesn't let you decide/count Y in general." That's what
| formalism is: a handy way of making statements about what
| you _can 't_ do constructively in the general case.
|
| To be fair, constructivists tend to prefer talk about
| different "universes" as opposed to different "sizes" of
| sets, but that's all it is: little more than a mere
| difference in terminology! You can show equiconsistency
| statements across these different points of view.
| btilly wrote:
| Yes, you can show such equiconsistency statements. As
| Godel proved, for any set of classical axioms, there is a
| corresponding set of intuitionistic axioms. And if the
| classical axioms are inconsistent, then so is the
| intuitionistic equivalent. (Given that intuitionistic
| reasoning is classically valid, an inconsistency in the
| intuitionistic axioms trivially gives you one in the
| classical axioms.)
|
| So the care that intuitionists take does not lead to any
| improvement in consistency.
|
| However the two approaches lead to very different notions
| of what it means for something to mathematically exist.
| Despite the formal correspondences, they lead to very
| different concepts of mathematics.
|
| I'm firmly of the belief that constructivism leads to
| concepts of existence that better fit the lay public than
| formalism does.
| edanm wrote:
| What leaps are "unprovable"? I'm curious, that doesn't sound
| right.
|
| For sure there are valid arguments on whether or not to use
| certain axioms which allow or disallow some set theoretical
| constructions, but given ZFC, is there anything that follows
| that is unprovable?
| btilly wrote:
| When you say "given ZFC", you're assuming a lot. Including
| a notion of mathematical existence which bears little
| relation to any concept that most lay people have of what
| mathematical existence might mean.
|
| In particular, you have made sufficient assumptions to
| prove that almost all real numbers that exist can never be
| specified in any possible finite description. In what sense
| do they exist? You also wind up with weirder things. Such
| as well-specified finite problems that provably have a
| polynomial time algorithm to solve...but for which it is
| impossible to find or verify that algorithm, or put an
| upper bound on the constants in the algorithm. In what
| sense does that algorithm exist, and is finite?
|
| Does that sound impossible? An example of an open problem
| whose algorithm may have those characteristics is an
| algorithm to decide which graphs can be drawn on a torus
| without any self-crossings.
|
| If our notion of "exists" is "constructable", all possible
| mathematical things can fit inside of a countable universe.
| No set can have more than that.
| tempfile wrote:
| You are being very cryptic. Are you trying to say that
| the existence of uncountable sets requires the axiom of
| choice? If you are, that's false. If you aren't, I'm not
| sure what you are trying to say.
| ogogmad wrote:
| He never mentioned the Axiom of Choice. I think he
| articulated his opinion clearly enough. It's his own
| subjective value judgement.
| btilly wrote:
| I'm definitely not trying to say that the existence of
| uncountable sets requires the axiom of choice. Cantor's
| diagonalization argument for the reals demonstrates
| otherwise.
|
| I'm saying that to go from the uncountability of the
| reals to the idea that this implies that the infinity of
| the reals is larger, requires making some important
| philosophical assumptions. Constructivism demonstrates
| that uncountable need not mean more.
|
| On the algorithm example, you could have asked what I was
| referring to.
|
| The result that I was referencing follows from the https:
| //en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo...
| . The theorem says that any class of finite graphs which
| is closed under graph minors, must be completely
| characterized by a finite set of forbidden minors. Given
| that set of forbidden minors, we can construct a
| polynomial time test for membership in the class - just
| test each forbidden minor in turn.
|
| The problem is that the theorem is nonconstructive. While
| it classically proves that the set exists, it provides no
| way to find it. Worse yet, it can be proven that in
| general there is no way to find or verify the minimal
| solution. Or even to provide an upper bound on the number
| of forbidden minors that will be required.
|
| This need not hold in special cases. For example planar
| graphs are characterized by 2 forbidden minors.
|
| For the toroidal graphs, as
| https://en.wikipedia.org/wiki/Toroidal_graph will verify,
| the list of known forbidden minors currently has 17,523
| graphs. We have no idea how many more there will be. Nor
| do we have any reason to believe that it is possible to
| verify the complete list in ZFC. Therefore the polynomial
| time algorithm that Robinson-Seymour says must exist,
| does not seem to exist in any meaningful and useful way.
| Such as, for example, being findable or provably correct
| from ZFC.
| somecontext wrote:
| > you're assuming a lot. Including a notion of
| mathematical existence which bears little relation to any
| concept that most lay people have of what mathematical
| existence might mean.
|
| John Horton Conway:
|
| > It's a funny thing that happens with mathematicians.
| What's the ontology of mathematical things? How do they
| exist? In what sense do they exist? There's no doubt that
| they do exist but you can't poke and prod them except by
| thinking about them. It's quite astonishing and I still
| don't understand it, having been a mathematician all my
| life. How can things be there without actually being
| there? There's no doubt that 2 is there or 3 or the
| square root of omega. They're very real things. I still
| don't know the sense in which mathematical objects exist,
| but they do. Of course, it's hard to say in what sense a
| cat is out there, too, but we know it is, very
| definitely. Cats have a stubborn reality but maybe
| numbers are stubborner still. You can't push a cat in a
| direction it doesn't want to go. You can't do it with a
| number either.
| edanm wrote:
| > When you say "given ZFC", you're assuming a lot.
|
| Errr, I'm just assuming the axioms of ZFC. That's
| literally all I'm doing.
|
| > In what sense do [numbers that can't be finitely
| specified] exist?
|
| In the sense that we can describe rules that lead to
| them, and describe how to work with them.
|
| I understand that you're trying to tie the notion of
| "existence" to constructability, and that's fine. That's
| one way to play the game. Another is to use ZFC and be
| fine with "weird, unintuitive to laypeople" outcomes.
| Both are interesting and valid things to do IMO. I'm just
| not sure why one is obviously "better" or "more real" or
| something. At the end, it's all just coming up with rules
| and figuring out what comes out of them.
| btilly wrote:
| My point is that going from a lay understanding of
| mathematics to "just accept ZFC" means jumping past a
| variety of debatable philosophical points, and accepting
| a standard collection of answers to them. Mathematicians
| gloss over that.
| zozbot234 wrote:
| > In what sense do they exist?
|
| In the sense that all statements of non-constructive
| "existence" are made, viz. "you can't prove that they
| don't exist in the general case", so you are _allowed_ to
| work under the stronger assumption that they also exist
| constructively, without any contradiction resulting. That
| can certainly be useful in some applications.
| btilly wrote:
| Sure, we can choose to work in a set of axioms that says
| that there exists an oracle that can solve the Halting
| problem.
|
| But the fact that such systems don't create
| contradictions emphatically *DOES NOT* demonstrate the
| constructive existence of such an oracle. Doubly not
| given that in various usual constructivist systems, it is
| easily provable that nothing that exists can serve as
| such an oracle.
| zozbot234 wrote:
| > emphatically _DOES NOT_ demonstrate the constructive
| existence of such an oracle
|
| Of course, but it shows that you can _assume_ that such
| an oracle exists whenever you are working under
| additional conditions where the existence of such a
| "special case" oracle makes sense to you, even though you
| can't show its existence in the general case. This
| outlook generalizes to all non-constructive existence
| statements (and disjunctive statements, as appropriate).
| It's emphatically _not_ the same as constructive
| existence, but it can nonetheless be useful.
| btilly wrote:
| That's like asserting the existence of a bank account in
| my name with a billion dollars in it that I know nothing
| about.
|
| I won't ever be able to find a contradiction from that
| claim, because I have no way to find that bank account if
| it exists.
|
| But that argument also won't convince me that the bank
| account exists.
| zozbot234 wrote:
| That argument ought to convince you that there's a mere
| "possible world" where that bank account turns out to
| exist. Sometimes we are implicitly interested in these
| special-cased "possible worlds", even though they'll
| involve conditions that we aren't quite sure about. Non-
| constructive existence is nothing more than a handy way
| of talking about such things, compared to the
| constructively correct "it's not the case that the
| existence of X is always falsified".
| btilly wrote:
| It would be weird for a constructivist to be interested
| in a possible world that they don't believe exists.
|
| Theoretically possible? Sure. But the kinds of questions
| that lead you there are generally in opposition to the
| kinds of principles that lead someone to prefer
| constructivism.
| drdeca wrote:
| If such a system proved that the answer to some decidable
| question was x, when the actual answer was y, then the
| system would prove a contradiction. If the system doesn't
| prove a contradiction, then that situation doesn't
| happen, so you can trust its answers to decidable
| questions.
|
| If the only questions you accept as meaningful are the
| decidable ones, then you can trust its answers for all
| the questions you accept as meaningful and for which it
| has answers.
|
| Also, "provable that nothing that exists can serve as
| such an oracle" seems pretty presumptive about what
| things can exist? Shouldn't that be more like, "nothing
| which can be given in such-and-such way (essentially, no
| computable procedure) can be such an oracle"?
|
| Why treat it as axiomatic that nothing that isn't Turing-
| computable can exist? It seems unlikely that any finite
| physical object can compute any deterministic non-Turing-
| computable function (because it seems like state spaces
| for bounded regions of space have bounded dimension), but
| that's not something that should be a priori, I think.
|
| I guess it wouldn't really be verifiable if such a
| machine did exist, because we would have no way to
| confirm that it never errs? Ah, wait, no, maybe using the
| MIP* = RE result, maybe we could in principle use that to
| test it?
| btilly wrote:
| You're literally talking about how I should regard the
| hypothetical answers that might be produced by something
| that I think doesn't exist. There's a pretty clear case
| of putting the cart before the horse here.
|
| On being presumptive about what things can exist, that's
| the whole point of constructivism. Things only exist when
| you can _construct_ them.
|
| We start with things that everyone accepts, like the
| natural numbers. We add to that all of the mathematical
| entities that can be constructed from those things. This
| provides us with a closed and countable universe of
| possible mathematical entities. We have a pretty clear
| notion of what it means for something in this universe to
| exist. We cannot be convinced of the existence of
| anything that is outside of the universe without making
| extra philosophical assumptions. Philosophical
| assumptions of exactly the kind that constructivists do
| not like.
|
| This constructible universe includes a model of
| computation that fits Turing machines. But it does not
| contain the ability to describe or run any procedure that
| can't fit onto a Turing machine.
|
| Therefore an oracle to decide the Halting problem does
| not exist within the constructible universe. And so your
| ability to imagine such an oracle, won't convince a
| constructivist to accept its existence.
| zozbot234 wrote:
| You can think that something doesn't exist in the general
| case, while still allowing that it might exist in
| unspecified narrow cases where additional constraints
| could apply. For example, there might be algorithms that
| can decide the halting problem for some non-Turing
| complete class of programs. Being able to talk in full
| generality about how such special cases might work is the
| whole point of non-constructive reasoning. It's "non-
| constructive" in that it states "I'm not going to
| construct this _just yet_ ".
| Etherlord87 wrote:
| You don't need an implicit philosophical assumption, you just
| need to define what an infinity is and the comparison method.
| btilly wrote:
| Here's a hint. When someone makes a reference to something
| that was actively debated for decades, and you're not
| familiar with said debates, you should probably assume that
| you're missing some piece of relevant knowledge.
|
| https://plato.stanford.edu/entries/mathematics-
| constructive/ is one place that you could start filling in
| that gap.
| nextaccountic wrote:
| This looks like a philosophical stance in the philosophy of
| mathematics actually, and it's called formalism
| thadt wrote:
| Probably not. But this one time we had an argument and I made
| a statement along the lines of "I'm right, naturally." She
| went irrational. I lost the argument.
|
| QED
| btilly wrote:
| LOL
|
| If she laughs at that kind of thing, I can see why you
| married her.
| ogogmad wrote:
| It might be relevant to look at this: https://home.sandiego.e
| du/~shulman/papers/jmm2022-complement...
|
| Also this: https://arxiv.org/pdf/1212.6543
|
| Assuming you haven't looked at these already, of course.
| btilly wrote:
| I had already read the second. I'm not so enthused about
| the first.
| mensetmanusman wrote:
| Don't worry, we have only decided that there are two sizes of
| Infinitis- normal ones and really big ones.
| gnulinux wrote:
| 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 wrote:
| 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 wrote:
| 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
| bmitc wrote:
| What else is it supposed to do?
| doe88 wrote:
| 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 wrote:
| 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/
| Tazerenix wrote:
| >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-...). 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 wrote:
| > 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.
| tiahura wrote:
| Mathematics is a science of formal systems. Proofs are its
| experiments, axioms its assumptions. Both math and science
| test consistency--one internally, the other against nature.
| Different methods, same spirit of systematic inquiry.
| myrmidon wrote:
| You could turn the argument around and say that math must be
| a science because it builds on falsifiable hypotheses and
| makes testable predictions.
|
| In the end arguing about whether mathematics is a science or
| not makes no more sense than bickering about tomates being
| fruit; can be answered both yes and no using reasonable
| definitions.
| TimPC wrote:
| In general you aren't testing as an empiricist though, you
| are looking for a rational argument to prove or disprove
| something.
| Tazerenix wrote:
| The practical experience of doing mathematics is actually
| quite close to a natural science, even if the subject is
| technically a "formal science* according to the
| conventional meanings of the terms.
|
| Mathematicians actually do the same thing as scientists:
| hypothesis building by extensive investigation of
| examples. Looking for examples which catch the boundary
| of established knowledge and try to break existing
| assumptions, etc. The difference comes after that in the
| nature of the concluding argument. A scientist performs
| experiments to validate or refute the hypothesis,
| establishing scientific proof (a kind of conditional or
| statistical truth required only to hold up to certain
| conditions, those upon which the claim was tested). A
| mathematician finds and writes a proof or creates a
| counter example.
|
| The failure of logical positivism and the rise of
| Popperian philosophy is obviously correct that we can't
| approach that end process in the natural sciences the way
| we do for maths, but the practical distinction between
| the subjects is not so clear.
|
| This is all without mention the much tighter coupling
| between the two modes of investigation at the boundary
| between maths and science in subjects like theoretical
| physics. There the line blurs almost completely and a
| major tool used by genuine physicists is literally
| purusiing mathematical consistency in their theories.
| This has been used to tremendous success (GR, Yang-Mills,
| the weak force) and with some difficulties (string
| theory).
|
| --------
|
| Einstein understood all this:
|
| > If, then, it is true that the axiomatic basis of
| theoretical physics cannot be extracted from experience
| but must be freely invented, can we ever hope to find the
| right way? Nay, more, has this right way any existence
| outside our illusions? Can we hope to be guided safely by
| experience at all when there exist theories (such as
| classical mechanics) which to a large extent do justice
| to experience, without getting to the root of the matter?
| I answer without hesitation that there is, in my opinion,
| a right way, and that we are capable of finding it. Our
| experience hitherto justifies us in believing that nature
| is the realisation of the simplest conceivable
| mathematical ideas. I am convinced that we can discover
| by means of purely mathematical constructions the
| concepts and the laws connecting them with each other,
| which furnish the key to the understanding of natural
| phenomena. Experience may suggest the appropriate
| mathematical concepts, but they most certainly cannot be
| deduced from it. Experience remains, of course, the sole
| criterion of the physical utility of a mathematical
| construction. But the creative principle resides in
| mathematics. In a certain sense, therefore, I hold it
| true that pure thought can grasp reality, as the ancients
| dreamed. - Albert Einstein
| 2snakes wrote:
| An alternative to abstraction is to use iconic forms and
| boundary math (containerization and void-based
| reasoning). See Laws of Form and William Bricken's books
| recently. Using a unary operator instead of binary
| (Boolean) does indeed seem simpler, in keeping with
| Nature. Introduction: https://www.frontiersin.org/journal
| s/psychology/articles/10....
| pdpi wrote:
| > In the end arguing about whether mathematics is a science
| or not makes no more sense than bickering about tomates
| being fruit
|
| That's the thing, though -- It does make sense, and it's an
| important distinction. There is a reason why "mathematical
| certainty" is an idiom -- we collectively understand that
| maths is in the business of irrefutable truths. I find that
| a large part of science skepticism comes from the
| fundamental misunderstanding that science is, like maths,
| in the business of irrefutable truths, when it is actually
| in the business of temporarily holding things as true until
| they're proven false. Because of this misunderstanding,
| skeptics assume that science being proven wrong is a
| deathblow to science itself instead of being an integral
| part of the process.
| The_suffocated wrote:
| Somewhat tangential to the discussion: I have once read that
| Richard Feynman was opposed to the idea (originally due to
| Karl Popper) that falsifiability is central to physics, but I
| haven't read any explanation.
| AlexandrB wrote:
| Mathematical "truth" all depends on what axioms you start
| with. So, in a sense, it doesn't prove "truth" either - just
| systemic consistency[1] given those starting axioms. Science
| at least grapples with observable phenomena in the universe.
|
| [1] And even _this_ has limits: https://en.wikipedia.org/wiki
| /Godel%27s_incompleteness_theor...
| GLdRH wrote:
| He probably means science in a wider sense as opposed to the
| anglo-american narrower sense where science is just physics,
| chemistry, biology and similar topics.
| weinzierl wrote:
| Pure mathematics is just symbol pushing and can never be
| science. It is lot of fun though and as it turned out
| occasionally pretty useful for science.
| lo_zamoyski wrote:
| It is absolutely a science, a formal science. What it isn't
| is an empirical science.
|
| The "symbol pushing" is a methodological tool, and a very
| useful one that opened up the possibility of new expansive
| fields of mathematics.
|
| (Of course, it is important to always distinguish between
| properties of the abstraction or the tool from the object
| of study.)
| weinzierl wrote:
| Well, we are talking about _pure_ mathematics and there
| is not much Popperian scientific method in it.
| Warwolt wrote:
| Who cares? That's just semantics. If we define science as
| the systematic search for truths, then mathematics and
| logic are the paradigmic sciences. If we define it as
| only empirical search for truth then perhaps that
| excludes mathematics, but it's an entirely unintersting
| point, since it says nothing.
| lo_zamoyski wrote:
| It's not an _empirical_ science, but it is a _science_ ,
| where "science" means any systematic body of knowledge of an
| aspect of a thing and its causes under a certain method. (In
| that sense, most of what are considered scientific fields are
| families of sciences.) Mathematics is what you'd call a
| _formal science_ with formal structure and quantity as its
| object of study and deductive inference and analysis as its
| primary methods (the cause of greatest interest is the formal
| cause).
| ubj wrote:
| Science involves both deductive and inductive reasoning. I
| would in turn argue that mathematics is a science that
| focuses heavily (but not entirely) on deductive reasoning.
| abdullahkhalids wrote:
| Mathematical proofs are checked by noisy finite computational
| machines (humans). Even computer proofs' inputs-outputs are
| interpreted by humans. Your uncertainty in a theorem is lower
| bounded by the inherent error rate of human brains.
| goatlover wrote:
| Plenty of mathematical proofs have been proven true with
| 100% certainty. Complicated proofs that involve a lot of
| steps and checking can have errors. They can also be proven
| true if exhaustively checked.
| naasking wrote:
| > Plenty of mathematical proofs have been proven true
| with 100% certainty
|
| Solipsists would like to have a word with you...
| drdeca wrote:
| This may be, but not, I think, in a way that is
| particularly worth modeling?
|
| When we try to model something probabilistically, it is
| usually not a great idea to model the probability that we
| made an error in our probability calculations as part of
| our calculations of the probability.
|
| Ultimately, we must act. It does no good to suppose that
| "perhaps all of our beliefs are incoherent and we are
| utterly incapable of reason".
| Aardwolf wrote:
| I'm not sure if it deals only with provable truths? It even
| deals with the concept of unprovability itself, if the
| incompleteness theorem is considered part of mathematics
| nitwit005 wrote:
| A proof is just an argument that something is true. Ideally,
| you've made an extremely strong argument, but it's still a
| human making a claim something is true. Plenty of published
| proofs have been shown to be false.
|
| Math is scientific in the sense that you've proposed a
| hypothesis, and others can test it.
| goatlover wrote:
| Difference is mathematical arguments can be shown to be
| provably true when exhaustively checked (which is straight
| forward with simpler proofs). Something you don't get with
| the empirical sciences.
|
| Also the empirical part means natural phenomena needs to be
| involved. Math can be purely abstract.
| nitwit005 wrote:
| You're making a strong argument if you believe you
| checked every possibility, but it's still just an
| argument.
|
| If you want to escape human fallibility, I'm afraid
| you're going to need divine intervention. Works checked
| as carefully as possible still seem to frequently feature
| corrections.
| ndriscoll wrote:
| Not only is intuition important (or the entire point; anyone
| with some basic training or even a computer can follow rules to
| do formal symbol manipulation. It's the intuition for what
| symbol manipulation to do when that's interesting), but it is
| literally discussed in a helpful, nonjudgmental way on Math
| Stack Exchange. e.g.
|
| https://math.stackexchange.com/questions/31859/what-concept-...
|
| Other great sources for quick intuition checks are Wikipedia
| and now LLMs, but mainly through putting in the work to
| discover the nuances that exist or learning related topics to
| develop that wider context for yourself.
| nkrisc wrote:
| > 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.
|
| I may be off-base as an outsider to mathematics, but Euclid's
| Elements, per my understanding, is very much grounded in the
| physical reality of the shapes and relationships he describes,
| if you were to physically construct them.
| empath75 wrote:
| Quite the opposite, Plato, several hundred years before
| Euclid was already talking about geometry as abstract, and
| indeed the world of ideas and mathematics as being _more
| real_ than the physical world, and Euclid is very much in
| that tradition.
|
| I am going to quote from the _very beginning_ of the
| elements:
|
| Definition 1. A point is that which has no part. Definition
| 2. A line is breadthless length.
|
| Both of these two definitions are impossible to construct
| physically right off the bat.
|
| All of the physically realized constructions of shapes were
| considered to basically be shadows of an idealized form of
| them.
| kannanvijayan wrote:
| Another point to keep in mind is that a lot of mathematics
| that's not considered abstract _now_ was definitely
| considered "hopelessly" abstract at the time of its
| conception.
|
| The complex number system started being explored by the
| greeks long before any notion of the value of complex
| spaces existed, and could be mapped to something in
| reality.
| mrguyorama wrote:
| Hell, 0 used to be considered too abstract!
| gaze wrote:
| The only things that are weird in math are things that would
| not be expected after understanding the definitions. A lot of
| the early hurdles in mathematics are just learning and gaining
| comfort with the fact that the object under scrutiny is nothing
| more than what it's defined to be.
| rob74 wrote:
| 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 wrote:
| > 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-
| architect...
|
| [2] https://en.wikipedia.org/wiki/Hyperuranion
| taeric wrote:
| I think it is fair to say that it was always an abstraction.
| But, crucially, it was built on language as much as it was
| empiricism.
|
| That is, numbers were specifically used to abstract over how
| other things behave using simple and strict rules. No?
| TuringTest wrote:
| > That is, numbers were specifically used to abstract over
| how other things behave using simple and strict rules. No?
|
| Agree that math is built on language. But math is not any
| specific set of abstractions; time and again mathematicians
| have found out that if you change the definitions and
| axioms, you achieve a quite different set of abstractions
| (different numbers, geometries, infinity sets...). Does it
| mean that the previous math ceases to exist when you find a
| contradiction on it? No, it's just that you start talking
| about new objects, because you have gained new knowledge.
|
| The math is not in the specific objects you find, it's in
| the process to find them. Rationalism consider on thinking
| one step at a time with rigor. Math is the language by
| which you explain rational thought in a very precise,
| unambiguous way. You can express many different thoughts,
| even inconsistent ones, with the same precise language of
| mathematics.
| taeric wrote:
| Agreed that we grew math to be that way. But there is an
| easy to trace history on the names of the numbers. Reals,
| Rationals, Imaginary, etc. They were largely named based
| on their relation to the language on how to relate them
| to physical things.
| stonemetal12 wrote:
| Mathematics arose from ancient humans need to count and
| measure. Even the invention\discovery of Calculus was in
| service to physics. It has probably only been 300 years or so
| since Mathematics has been symbolic, before that it was more
| geometric and more attached to the physical world.
|
| Leibniz (late 1600s) helped to popularize negative numbers. At
| the time most mathematicians thought they were "absurd" and
| "fictitious".
|
| No, not highly abstract from the beginning.
| elliotec wrote:
| Sorry what? Ancient humans invented symbols to count. How is
| that not symbolic?
|
| Geometry is "attached" to the physical world... but in an
| abstract way... but you can point to the thing your measuring
| maybe so it doesn't count...
|
| Abstraction was perfected if not invented by mathematics.
| Ekaros wrote:
| Symbolic here refers of doing math with place holders, be
| it letters or something. Ancient world had notations for
| recording numbers. But much less so to do math with them.
| Say like long division.
| empath75 wrote:
| Almost from the first time people started writing about
| mathematics, they were writing about it in an abstract way.
| The Egyptians and the Babylonians kept things relatively
| concrete and mostly stuck to word problems (although lists of
| pythagorean triples is evidence for very early "number
| theory"), but Greece, China and India were all working in
| abstractions relatively early.
| hollerith wrote:
| In particular, ancient Greek geometry at least after 300 BC
| proceeded from axioms, which is a central component of the
| abstract approach.
| compressedgas wrote:
| > Leibniz (late 1600s) helped to popularize negative numbers.
|
| Wasn't that imaginary numbers?
| anthk wrote:
| Archimedes did Calculus before Newton.
|
| https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theor.
| ..
| lo_zamoyski wrote:
| It is abstract in the strict sense, of course. Every science
| is, as "abstract" simply means "not concrete". All reasoning is
| by definition abstract in the sense it all reasoning by
| definition involved concepts, and concepts are by definition
| abstract.
|
| Numbers, for example, are abstract in the sense that you cannot
| find concrete numbers walking around or falling off trees or
| whatever. They're quantities abstracted from concrete
| particulars.
|
| What the author is concerned with is how mathematics became
| _so_ abstract.
|
| You have _abstractions_ that bear no apparent relation to
| concrete reality, at least not according to any direct
| correspondence. You have _degrees of abstraction_ that
| generalize various fields of mathematics in a way that are
| increasingly far removed from concrete reality.
| elliotec wrote:
| Right? Math is abstraction at its very core. Ridiculous premise
| acting as if this is anything but beyond ancient.
| jjgreen wrote:
| The number 1 is what a cow, a fox, a stone ... have in common,
| oneness. Mathematics _is_ abstraction, written down.
| prmph wrote:
| 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 wrote:
| 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 wrote:
| There was a time, not that long ago in human history, that zero
| was "so abstract".
| dist-epoch wrote:
| It was a religious offense to talk about zero.
|
| https://cambriamathtutors.com/zero-christianity/
| stonemetal12 wrote:
| 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 wrote:
| Unlike Zeno's famous example the paradox which does better at
| explaining the problem is
| 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_ wrote:
| Theirs no such thing as excessive abstraction in math, because
| abstraction is the point. Is category theory "excessive
| abstraction" in your opinion?
| fidotron wrote:
| > 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 wrote:
| > 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 wrote:
| I love how you lot just redefine words to suit your
| purpose:
|
| 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.
| empath75 wrote:
| I am not, this is what formal logic and formal reasoning
| means:
|
| https://plato.stanford.edu/entries/logic-classical/
|
| "Formal" in logic has a very precise technical meaning.
| fidotron wrote:
| What you mean is someone has redefined the word to suit
| their purpose, which is precisely what I pointed out at
| the top.
|
| Edit to add: this comment had a sibling, that was
| suggesting that given a specific proof assistant requires
| all input to be formal logic perhaps the word formal
| could be redefined to mean that which is accepted by the
| proof assistant. Sadly this fine example of my point has
| been deleted.
| hollerith wrote:
| Every mathematician understands what a formal proof is.
| Ditto a formal statement of a mathematical or logical
| proposition. The mathematicians of 100 years ago also all
| understood, and the meaning hasn't changed over the 100
| years.
| fidotron wrote:
| > The mathematicians of 100 years ago also all
| understood, and the meaning hasn't changed over the 100
| years.
|
| Isn't that the subject of the whole argument? That
| mathematicians have taken the road off in a very specific
| direction, and everyone disagreeing is ejected from the
| field, rather like occurred more recently in theoretical
| physics with string theory.
|
| Prior to that time quite clearly you had formal proofs
| which do not meet the symbolic abstraction requirements
| that pure mathematicians apparently believe are axiomatic
| to their field today, even if they attempt to pretend
| otherwise, as argued over the case of Euclid elsewhere.
| If the Pythagoreans were reincarnated, as they probably
| expected, they would no doubt be dismissed as crackpots
| by these same people.
| hollerith wrote:
| >quite clearly you had formal proofs which do not meet
| the symbolic abstraction requirements
|
| I've been unable to imagine or recall an example. Can you
| provide one?
| azan_ wrote:
| No, abstraction is the point and formal reasoning is a
| tool. And yes, what Euclid did is obviously abstraction, I
| don't know why so you consider this stance nonsense.
| fidotron wrote:
| Can you say how mathematics is inherently abstract in a
| way consistent with your day-to-day life as a concrete
| person? Or is your personhood also an abstraction?
|
| I could construct a formal reasoning scheme involving
| rules and jugs on my table, where we can pour liquids
| from one to another. It would be in no way symbolic,
| since it could use the liquids directly to simply be what
| they are. Is constructing and studing such a mechanism
| not mathematics? Similarly with something like musical
| intervals.
| azan_ wrote:
| Of course I can. I frequently use numbers which are great
| abstraction. I can use same number five to describe
| apples, bananas and everything countable.
| fidotron wrote:
| > to describe apples, bananas and everything countable
|
| An apple is an abstraction over the particles/waves that
| comprise it, as is a banana.
|
| Euclid is no more abstract than the day to day existence
| of a normal person, hence to claim that it is unusually
| abstract is to ignore, as you did, the abstraction
| inherent in day to day life.
|
| As I pointed out it's very possible to create formal
| reasoning systems which are not symbolic or abstract, but
| due to that are we to assume constructing or studying
| them would not be a mathematical exercise? In fact the
| Pythagoreans did all sorts of stuff like that.
| azan_ wrote:
| > An apple is an abstraction over the particles/waves
| that comprise it, as is a banana.
|
| No, you don't understand what abstraction is. Apple is
| exactly arrangement of particles, it's not abstraction
| over them.
|
| > hence to claim that it is unusually abstract
|
| Who talks about him being unusually abstract (and not
| just abstract)?
|
| > is to ignore, as you did, the abstraction inherent in
| day to day life.
|
| How am I ignoring this abstraction when I've provided you
| exactly that (numbers are abstraction inherent in day to
| day life). I'm sorry but you seem to be discussing in bad
| faith.
| fidotron wrote:
| > Apple is exactly arrangement of particles, it's not
| abstraction over them.
|
| No. You can do things to that apple, such as bite it, and
| it is still an apple, despite it now having a different
| set of particles. It is the abstract concept of appleness
| (which we define . . . somehow) applied to that
| arrangement of particles.
|
| > I'm sorry but you seem to be discussing in bad faith.
|
| Really?
|
| > No, you don't understand what abstraction is.
| OkayPhysicist wrote:
| Complaining about jargon is lazy. Most communications about
| complicated things are not aimed at the layman, because to do
| anything useful with the complicated things, you tend to have
| to understand a fair amount of the context of the field. Once
| you're committed to actually learning about the field, the
| jargon is the easiest part: they're just words or phrases that
| mean something very specific.
|
| To put it another way: Jargon is the source code of the
| sciences. To an outsider, looking in on software development,
| they see the somewhat impenetrable wall of parentheses and
| semicolons and go "Ah, that's why programming is hard: you have
| to understand code". And I hope everyone here can understand
| that that's an uninformed thing to say. Syntax is the easy part
| of programming, it was made specifically to make expressing the
| rigorous problem solving easier. Jargon is the same way: it
| exists to make expressing very specific things that only people
| in this subfield actually think about easier, instead of having
| to vaguely gesture at the concept, or completely redefine it
| every time anybody wants to communicate within the field.
| ndriscoll wrote:
| Abstraction isn't to gatekeep; it's to increase the utility.
| It's the same as "dependency inversion" in programming: do your
| logic in terms of interfaces/properties, not in terms of a
| particular instance. This makes reasoning reusable. It also
| often makes things _clearer_ by cutting out distracting details
| that aren 't related to the core idea.
|
| People are aware that you need context to motivate
| abstractions. That's why we start with numbers and fractions
| and not ideals and localizations.
|
| Jargon in any field is to communicate quickly with precision.
| Again the point is not to gatekeep. It's that e.g. doctors
| spend a lot of time talking to other doctors about complex
| medical topics, and need a high bandwidth way to discuss things
| that may require a lot of nuance. The gatekeeping is not about
| knowing the words; it's knowing all of the information that the
| words are condensing.
| elAhmo wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| aristofun wrote:
| 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 wrote:
| > 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.
| s20n wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| I like Peano, but he was using Grassmann's definition of natural
| numbers
| tphyahoo2 wrote:
| Just drop the axiom of infinity and quit whining.
|
| https://en.wikipedia.org/wiki/Ultrafinitism
| ogogmad wrote:
| "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 wrote:
| 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.
| trinsic2 wrote:
| >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 wrote:
| 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 wrote:
| 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.
| lottin wrote:
| I wish the scroll bar was a little less invisible.
| BrandoElFollito wrote:
| 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 wrote:
| This article can also be written as "The unreasonable
| effectiveness of abstraction in mathematics."
| Animats wrote:
| 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.
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