21 ms·

What Is Math?

133 points by optimalsolver 6y ago

Pick-A-Hill2019 6y ago
An excellent article/submission, one of those huh, never thought to even think about that moments. I especially enjoyed the replies by Eugenia Cheng [1] on points 3 and 7 (from a link in the submitted article). [1] http://eugeniacheng.com/wp-content/uploads/2020/08/gracie-twitter.pdf http://eugeniacheng.com/wp-content/uploads/2020/08/gracie-tw...
loopz 6y ago
We have taxonomies of objects and concepts, but do we have taxonomy of relationships? This might continue to mystify us as long as we try to push the square block into the round hole. It worked so well with the smaller pieces.. Every new discovery started with a question and letting go of some assertions.
- mceldeen 6y ago
  Category Theory might be a good fit for this. There’s even a category of morphisms (relationships). Perhaps I’m misunderstanding your comment.
- aliceryhl 6y ago
  Yes. The book Thinking about mathematics by Stewart Shapiro covers it in chapter 10 and calls it Structuralism. https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_m...
ducaale 6y ago
A video which discusses this subject and compares different schools of thought https://youtu.be/AmySxYHqQCQ https://youtu.be/AmySxYHqQCQ
- cloogshicer 6y ago
  Came here to post this. It's absolutely wonderful!
- 3np 6y ago
  https://www.youtube.com/watch?v=Kh0Y2hVe_bw https://www.youtube.com/watch?v=Kh0Y2hVe_bw Or, more on-topic from the same: https://vimeo.com/77451201 https://vimeo.com/77451201
  - Quekid5 6y ago
    Oh god, I love that Look Around You (What are...) skit so much. It's just so incredibly silly.
- ImaCake 6y ago
  I particularly like the closing comments about how we teach math. It is something of a theme that we do a bad job of explaining why math should be taught. It took me many years in academic research to realise that not only was math useful to me, but that I liked it. If we could do a better job communicating what math can do - make us rich, predict the future, win at games - maybe it would hold the attention of our students better.
- marc_abonce 6y ago
  I'm not even epileptic, but that video felt like it would give me a seizure at any moment.
willis936 6y ago
It’s important to note that no model using math has accurately represented nature. Just because simple approximations line up in a beautiful way is not reason to think they are 1:1 the same. Math is not sacred or divine. It’s a tool. Like all tools: its shape matters. Math, as we know it, is invented. It is one set of rules that we like to use to describe things. It is not the only way. I can imagine an intelligent species that has a structured set of logical rules that looks very different from our math but still be able to make useful models.
- senthil_rajasek 6y ago
  "It’s important to note that no model using math has accurately represented nature" (Edit: I am getting downvoted, so let me expand my answer a bit. Nature is explained using Physics and Physics uses Mathematics for its models of Nature.) This is part of Godel's discovery. Especially, his first Incompleteness Theorem. https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes...
  - tsimionescu 6y ago
    The incompleteness theorems don't have anything to do with nature, they are about pure mathematics.
    - senthil_rajasek 6y ago
      Physics explains nature using mathematics.
      - protonfish 6y ago
        And writers explain nature using words.
        senthil_rajasek 6y ago
        Not as precisely as mathematics can as shown by F = ma or E = mc^2.
spodek 6y ago
One could equally ask what is history, art, sports, etc. The answers mathematicians give seem partly interesting, but also, to me, imply insecurity -- that they have to justify to others that everyone should learn it. Not many other subjects suggest needing justification. Meanwhile, kids teach themselves math in self-directed learning environments https://www.psychologytoday.com/us/blog/freedom-learn/201004/kids-learn-math-easily-when-they-control-their-own-learning https://www.psychologytoday.com/us/blog/freedom-learn/201004.... Up to high school they taught it to me by textbook, mainly teachers who didn't know or care about it. I found beauty in it and loved it despite them. I suspect if we taught it less relying on authority and grades, more engaging and based on curiosity, we'd appreciate it more. Instead of asking what it is, we'd do it. Even people who don't play music don't ask what it is, like it doesn't need to exist. They probably would if we just taught music theory from textbooks but not singing and dancing.
- mcphage 6y ago
  > One could equally ask what is history, art, sports, etc. Yes, those are all meaningful, interesting questions that people have pondered for a very long time.
- TheBigSalad 6y ago
  Kids just have no motivation for math. It's not obviously enjoyable like art and music. Teaching what it is and how it helps them in the real world motivates them to learn.
  - jng 6y ago
    Some kids do (I did).
aliceryhl 6y ago
Platonism isn't the only way to belive that mathematical truth exists beyond humans. In Platonism one believes that the objects themselves are real in some sense, but I prefer to see it in a different light: mathematics is about considering the relationships between arbitrary objects. The actual objects don't matter - what matter is how they are connected, and the things we prove are true whenever a collection of objects fit together with that relationship.
- proc0 6y ago
  > mathematics is about considering the relationships between arbitrary objects. The actual objects don't matter - what matter is how they are connected You know this is Category Theory I assume? I guess it does describe math abstractly, but in a way it is also math. Not sure tbh.
wruza 6y ago
Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all? How does one find ethical or religious problems and questions in pure logic? Logic is fundamental and proven to be able to apply in reality. Ethics and religion operate over multiple levels of qualitative transitions and cannot(?) be fundamental. Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?) Because DoJC exists outside of space and time in minds of believers? It just doesn't have a physical counterpart and does barely have any logic to evolve/deduct from it, for us to predict events based on DoJC as if it was applicable to parts of reality. It is understandable how someone could ask "stupid" (not really) questions about maths, but when philosophers start to draw parallels, it is easy to lose track of what they are even talking about. Can someone explain what they meant exactly?
- koheripbal 6y ago
  There are some core questions in ethics that might be considered so fundamental that they almost don't seem like religious question at all. The Observer Problem is a good example. I experience life from this singular human's perspective. Why this specific one? What mechanism dictates which human my "observer" occupies? Why even a human and not an animal? The word "soul" is a poor descriptor because it is associated with so much more, but the Observer Problem is real - or at least it might be. I heard a theory is that there is another higher order time dimension, and all of our "observers" are actually the same one living in some sequence "simultaneously". ...but that doesnt really solve the dilemma either because even that sequence would need some mysterious ordering mechanism. ...but in all that there are core questions about what is time, what is space, what is energy, what happens to the universe... etc One thing I tell me daughters is that death isn't scary. ...because we exist inna limited space and time. We should be no sadder to not exist in the future than not existing 10m to the left. We cannot be eternal, but if you recognize that time is just a dimension, then our existence is permanant in the history of the universe. Nothing can undo what we did and who we were.
ahelwer 6y ago
I've always liked the idea of math as a type of language. When you learn quantum computing, for example, you go beyond the limits of our informal language (English, etc.) which developed in a purely classical world. This is why pop science articles on quantum phenomena never really make any sense. They try to use metaphors to explain concepts, but there is no existing concept in our informal language to use as a referent. Many quantum phenomena belong to an ontological category that is entirely new to most adults. For example, superposition: you try to explain it as something being in two places at once, but this doesn't make sense nor is it very helpful for understanding how objects in superposition behave. In order to understand quantum concepts you have to learn a new language - that of mathematics. Then it becomes easy! Superposition is just linear combination. Done. Also if you enjoy thinking about this topic I recommend Neal Stephenson's book Anathem, which I just finished. It is a complete slog at the start, with endless boring architectural descriptions. I listened to it in audiobook form which made that section much more pleasant.
- dfischer 6y ago
  In my understanding, that’s exactly what math is, a language. And various modes of thought have utilized the language of mathematics to communicate ideas that were otherwise not possible or not consistently able to arrive at same conclusions. This is very interesting. What is language? When does language evolve? Is language itself consciousness? Whenever there’s a leap in language is it because something had to be constructed to help assign an abstraction to a coordinate in space where thoughts exist? It appears we’re languages all the way down. Or analogicalmachines? Lol. Lots to unpack here - some with what many would call not nice Consequences. What happens when a minority elite (unsure how else to describe it?) disciplines require a language that the majority lack the finesse to use? Example: math used in physics, or programming in computation. If the languages being used introduce more complexity without the ability to reduce net complexity then it damages the communication of the specie using it. I’ve been thinking about this quite a bit on our trajectory as a species. Complexity in language leads to information silos which leads to various existential risks based on information and miscommunication. Also interesting coercion / authoritarian emergence comes out of those creating a new language to reason about an abstraction which leads to a success that negatively impacts those which do not know the language and if the space to understand said language is sufficiently far enough you have much room for manipulation. Also, who gets to say which language is the right one? What happens when a new language is created that is more advanced and the majority disallow it preventing an evolution? Interesting dynamics.
gumby 6y ago
She asked two important questions, only one of which (the epistemology of mathematics)was addressed in the article. The other one was “why were the Greeks (and contemporaneous Indians) wondering about geometry of a plane and why were the Arabs thinking of algebra” when they had more limited technology than we do?
- naringas 6y ago
  maybe because they distributed the output of their limited technology better than we do? there's just too much busy-work in modernity
- TeMPOraL 6y ago
  The latter was covered in the answers by Eugenia Cheng, linked in the article: http://eugeniacheng.com/wp-content/uploads/2020/08/gracie-twitter.pdf http://eugeniacheng.com/wp-content/uploads/2020/08/gracie-tw.... TL;DR: they were curious, they saw some patterns in their daily experiences, and tried to generalize them.
JohnStrangeII 6y ago
This is a rather interesting article. I'm a Platonist. From what we know today, real numbers cannot exist in a finite space, but they seem to exist mathematically. The same can be said about many other mathematical structures, including those that can only be characterized adequately (categorically) in higher-order logic. It's also worth noting that physical and mathematical existence are based on completely different criteria. For non-constructionists mathematical objects exist once they are not demonstrably contradictory (although the absence of contradictions often cannot be proved in an absolute sense). In contrast to this, for physicists an object exists once it can be measured, where measurement is ultimately tied to sensual experience. There are also theoretical entities in physics that cannot be directly measured, but their existence is usually downplayed, they're not supposed to "really" exist but only as theory-dependent entities. In any case,the two "kinds of existence" are very different from each other.
- dfischer 6y ago
  Quite an astute observation. I’ve been thinking about this quite heavily recently. It has odd occult references too in the sense of “nothing creates something.” That which is measured, exists. No measurement itself, as in of itself as a thing, is a thing that exists. You can measure 3cm but “3” and “cm” don’t exist. Virtual values assigned by consensus. All axioms are agreements. Truth or existence itself is convergent and resembles certainty only in the majority agreeing to it. Imagination allows nothingness (imaginary things) to be measured. Imagine any unit, give it a “rule” and now it can be measured! Pretty interesting bridge between the “imaginary” and the “real.” There’s so much nuance on what existence is, and isn’t, or even not is nor isn’t is depending on if you subscribe to classical logic based on what we currently describe as mostly Aristolian or if you use many-valued logics which are making a come back yet rooted in the Vedas... I find this extremely profound and it’s one of my top focal areas of study right now. It’s also related to my observation of disagreement, chaos, and behavior of the human animal. It seems the zeitgeist of the egregore lacks the ability to adjust perspective and see truth depending on the “rule/roles being played” (all disciplines are games/acting in a way). Tons to unpack.
- TheOtherHobbes 6y ago
  The perception of numbers exists mathematically. Likewise for more abstract structures. You cannot show me threeness. You can show three things and tell me to generalise, and if you keep generalising you'll end up with a consistent-ish framework of sorts for your observations that can be applied to certain other physical experiences. But fundamentally this is an exploration of the consequences of psychological processes - like perceptual grouping, and inductive relationship inference - not observations of external phenomena. It doesn't seem that way, but there is no external authority you can appeal to which will state definitively that when mathematicians all agree on something their experience of "true" is absolutely and objectively correct, and not a distorted and limited artefact of human cognition.
delhanty 6y ago
>Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist ... That statement surprised me! Google told me that according to this biography by J J O'Connor and E F Robertson [0], Roger Penrose in a debate with Stephen Hawking in 1994 self-described as a realist rather than a Platonist. They quote Penrose as follows: >At the beginning of this debate Stephen said that he thinks that he is a positivist, whereas I am a Platonist. I am happy with him being a positivist, but I think that the crucial point here is, rather, that I am a realist. Also, if one compares this debate with the famous debate of Bohr and Einstein, some seventy years ago, I should think that Stephen plays the role of Bohr, whereas I play Einstein's role! For Einstein argued that there should exist something like a real world, not necessarily represented by a wave function, whereas Bohr stressed that the wave function doesn't describe a "real" microworld but only "knowledge" that is useful for making predictions. [0] https://mathshistory.st-andrews.ac.uk/Biographies/Penrose/ https://mathshistory.st-andrews.ac.uk/Biographies/Penrose/
thinkloop 6y ago
Math is a language that describes relationships between quantities of different things.
jng 6y ago
I like to think about it as this: numbers are a human invention, but quantity is a property of the universe. The laws of physics, including all types of conservation, take care of that. Quantity takes care of ensuring many derived properties (rules) that numbers show and which can be expressed abstractly (complex math). Studying the properties of a complex self-consistent system is interesting, and sometimes, when it reveals an interesting resulting property, it can be traced all the way back to the original natural property of quantity, and applied in a practical way in the world. Other times, it can't, and it remains a curious interesting fact in the realm of the abstract system.
- hatch_q 6y ago
  I did a thought experiment few years back: Is anything real if we live in a simulation? I came to conclusion that all physical laws, chemistry, the universe would be simulated. And the only "truth" that is the same for host and simulation is math. Imagine that you're creating computer game - a limited simulation. You can change laws of physics, chemistry, life, etc... but math will always hold true. 1 + 1 will always be 2.
  - cyberbanjo 6y ago
    Reminds me of Geohotz Breaking the Simulation.
082349872349872 6y ago
The idea of something existing “outside of space and time” makes empiricists nervous This seems like a boring objection: where does the letter 'A', or the ascii 0x41, exist? I'm willing to admit that concrete instances of both exist in space and time (and I might even admit maps forming a galois connection between those and the abstract ideal 'A') but there's no "reference" canonical A, located in time, located in space, giving them meaning.
- codegladiator 6y ago
  > This seems like a boring objection: where does the letter 'A', or the ascii 0x41, exist? Only in our minds ? A doesn't really exist IMO.
deleted 6y ago
[deleted]
thenoblesunfish 6y ago
One could argue that a defining characteristic of human intelligence is our ability to agree on "unreal" concepts. I think the first time I heard it explicitly was in the book "Sapiens", though I doubt it originated there. In this sense, mathematical concepts are no different than, say Santa Claus. We can agree (mostly) about what it is we're talking about, and talk about it. What sets mathematical "unreal" ideas apart is just how useful they seem to be in describing the universe. They're patterns that fit onto lots and lots of things. For instance, we can all sort of agree on the idea of a (finite) set, and on some basic rules of logic, and with those you can build a lot of mathematics.
dfischer 6y ago
(Commented on someone but they deleted. Will post as parent? All axioms are imaginary. Given that humanities method of logic is imaginary, then consensus is the only difference between certainty and non certainty which is equivalent to truth. Logic is programming. Human’s are programmers and hardware/OS who largely don’t realize they’re running arbitrary truth values based on popular imaginary rules. To exist, is to LARP? Lol.
nendroid 6y ago
Whether math is like an art or whether it is "discovered" is sort of a pointless question to me. To start off, math is simply the art of understanding the consequences and theorems from a set of axioms. Since there are an infinite amount of sets of axioms, there's an infinite amount of math. Whether a set of axioms or the theorems derived from a set of axioms is discovered or created is again a pointless philosophical debate that is not exclusive to mathematics. You can apply this question to any topic. Case in point: There is a finite set of paintings that can be painted on a fixed sized canvas using a fixed set of colored paint. When you create a painting... are you creating a painting or are you "discovering it" from the set of all paintings? Since the question does not pertain exclusively to math; if you think hard you can sort of realize that it's just philosophical speculation derived from intricacies in our language. Is anything truly created? Or is it just "discovered" from the set of all possible things? When we think about this stuff we're not actually thinking about anything profound, rather we are just examining intricacies of language and the definition of the word "discover" and "create." I mean if you think really hard and try to formalize the definition of "create" it can be said that to "create" something is to "discover" a single member from the set of all things. It can also be said to "discover" something is to "create" a new member from the set of all things. People are just getting excited over a language issue. The word "create" was never formalized and forms a sort of fuzzy overlap with "discover." Basically it's one and the same thing sort of like is the glass half full or half empty? The actual reality of this problem is uninteresting and pointless. This paradoxical-like phenomenon doesn't actually exist and arises exclusively because in our language, we have a word called "empty" and we have a word called "full." Consider the black wall. Is the wall "black" or does it lack color? Consider the flying bird. Is the bird flying or is it falling and missing the ground? Is the dark room full of darkness or is light just absent? It's all just language tricks I can "create" or "discover" an infinite amount of garbage here for you to think "profoundly" about without actually realizing it's just a language issue.
- tim333 6y ago
  Dunno - math seems to underlie the laws of physics and hence existence itself which is not just a language thing.
euske 6y ago
I like the idea of math being a fiction, but the difference between math and novel is that in math you can never ever break the rules once they're set. In contrast, I see that novelists constantly ignore or retrofit the rules to make it interesting. Imagine this was allowed in math! I also think that the fascinating part of programming is that it's sitting in between math and fiction, depending on an application. Sometimes you have to obey the rules very strictly (e.g. writing a parser), but in other times you can be wildly imaginative (e.g. designing UI).
- 082349872349872 6y ago
  Non-euclidean geometry was a retcon.
- ahelwer 6y ago
  Well, people certainly invent novel proof techniques which remain controversial for over a century. For example, Cantor's diagonal argument, which people still deny the validity of today (although those who do are generally considered cranks).
motohagiography 6y ago
In fewer words than the article, there's a concept called a substrate, kind of like what water in an oocean is to a fish, and while we experience the patterns of it all the time as part of our normal existence, expressing it in language that lets us share our experiences of it with each other is very fiddly. But when we manage to describe things that consistently reflect other peoples experiences, it is like drawing a map. You can experience the territory without it, but it's a totally different game when you have one. So between the critical theory poles of empiricism and Platonism is our ability to perceive and express our experience of our substrate. (What is our substrate? Turtles, all the way down.) Institutions these days never seem to pass up a teachable moment when it comes to reinforcing the precepts of dialectical materialism. e.g. "If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)" It's so tedious as to make me doubt their humanity, but I'm glad this girl sparked a popular discussion.
konjin 6y ago
Perhaps a history lesson would benefit her more than one in mathematics. We don't know much about Pythagoras and why he did mathematics. But we do know a bit more about Archimedes and the reasons why he did mathematics: to kill Romans and improve plumbing. Two things that mathematics was found to be extremely good at.
afiori 6y ago
my favourite (less than insightful) answer is that we live in a (hard) magic world and math then is the language of magic in a literary sense, which can then be used to formulate spells in like physic, chemistry, or psychology (we too are magic and can interact with the magic source in our own special way)
pessimizer 6y ago
Math is the series of statements that can be made about things that are true. Math can't tell you that there are three of [anything that you encounter with your sensory organs], but it can tell you that if there are three of those, you can also say... In that way, it tells you about the real world while having no connection to any real world. The problem is that as soon as a math-predicted observation fails, we can blame the mistake on our sensory organs - either formatively (when we assigned a particular observation to a particular mathematical entity (like "three"), or conclusively (when we assigned a mathematical result to a sensation.) If a mathematical result consistently fails to apply to any sensation, it ceases to be math - not by disappearing, but by failing to be consistent; if you can't apply a mathematical process to a sensation in a way that enables you to predict a subsequent sensation, there's no way to determine whether the math makes any consistent statement. Math has to be empirical because it's completely vetted by sensational observations. If a hypothetical god changed some aspect of the underlying physical reality (inclusive of changing some aspect of human sensory organs), the math would change along with it, and things that were math would cease to be math. If you included that god within your mathematics, math becomes the set of all possible maths, which is just the universal set. There's no meaning to a math that says everything about everything. I guess you can say Platonism is true too, since math is a statement about an underlying reality (because it is discarded when it is not), but it's not a meaningful thing to say. Math that turned out to be correct (and continues to turn out to be correct) is Platonic. Math that was incorrect (or in the future becomes incorrect) is not Platonic. The only way you can determine which math is Platonic and which math is not is through empirical observation, and the degree to which you can be sure that particular math continues to be Platonic is through how well your past observations predict your future observations. To answer the tweet, people study math because because being able to predict things gives you an advantage in your work.
yantrams 6y ago
For anyone interested in the topics discussed here, I highly recommend What is Mathematics, Really? [1] by Reuben Hersh [1] https://www.amazon.com/What-Mathematics-Really-Reuben-Hersh/dp/0195130871/ https://www.amazon.com/What-Mathematics-Really-Reuben-Hersh/...
- tim333 6y ago
  I read that and was a little disappointed. He argues it's basically just a human invention and skirts around the obvious counter arguments that say aliens would discover the same maths if with different language to describe it. He also tries to politicise the debate which I feel is kind of silly.
ArtWomb 6y ago
This is wonderful. I love when memes breed philosophy. We're teaching math wrong. It's as plain as day to me. Or rather we teach the wrong maths to the wrong students. If Marcie can't find a useful application on day one of the "theory" she learns in school, what is the point? It's a 19th century curriculum for a 21st century world. Basic accounting and personal finance would probably serve most students better. As to the epistemological quandary, I always return to Geometry or Visual Mathematics. Consider the infinite ways of calculating the digits of PI. In "Playing Pool with PI", Galperin demonstrated an analog computer that could calculate the digits of PI using only the perfectly elastic collisions of two billiard balls on a pool table. It works for all integral base number systems, and can even be extended to irrational numbers. And recently Google Brain's Adam Brown made an equally stunning observation: Galperin's bouncing billiard balls look identical to the Grover Quantum Search algorithm! Playing Pool with |ψ⟩: from Bouncing Billiards to Quantum Search https://arxiv.org/abs/1912.02207 https://arxiv.org/abs/1912.02207
- yeellow 6y ago
  > If Marcie can't find a useful application on day one of the "theory" she learns in school, what is the point? It's a 19th century curriculum for a 21st century world. Basic accounting and personal finance would probably serve most students better. Sorry, but following this logic we will end up with a conclusion that almost no school knowledge is "useful". How useful is playing volleyball? Reading poetry? Painting? Studying history? Biology would be reduced to first aid and basic medicine. I have those discussions with my daughters who learn how to do long multiplication. How is it useful if even I who know how to do it would rather use a calculator on my mobile instead? We would reduce math to basic operations on a calculators, probably nothing more would be considered useful by the majority of the population. The same apply to all other subjects.
  - unjurios 6y ago
    I think "application" is the wrong focus. The real thing to help students understand is this exact question: what is math? That remains a mystery to most students. It gets conflated with the problem of application, but I think the real problem is most educations don't make clear the idea of abstraction, symbol systems, and logic. When you don't understand something, especially why it might be useful or at least intrinsically interesting, it's hard to get motivated about it.
cskinner 6y ago
The article spends most of its words discussing the reality of math via Platonism vs empiricism while forgetting its own title "What is Math?" If going for a pithy answer, I like tha following quote by G. H. Hardy: A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. I think I originally saw this quote in "A Mathematician's Lament" [0] (also known by the author's name as "Lockhart's Lament"), which has a lot to say about modern math education. [0] (PDF) https://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf https://www.maa.org/sites/default/files/pdf/devlin/Lockharts...
roenxi 6y ago
What is interesting is I'm not sure how many commenters of the video listened to it. The actual points she raised were mostly focused on "why would Pythagoras have used algebra?". And the answer as far as I can see is he probably didn't. Most of the notation we use was introduced in the 1600s-1700s [0]. The concepts we are taught today, and the way we are taught them, would be foreign to Pythagoras. According to wiki, symbolic algebra only emerged in 1200+ with the Arabs (which probably links to why we don't use Greek by default for algebra but we do with geometry). [0]] https://en.wikipedia.org/wiki/History_of_mathematical_notation#Symbolic_stage https://en.wikipedia.org/wiki/History_of_mathematical_notati...
- tom_mellior 6y ago
  Pythagoras didn't use modern algebraic notation, but I don't think it's correct to conclude from this that what he was doing wasn't algebra. He reasoned about relationships between abstract quantities in ways that we today would write down in algebraic notation. Does a change of notation mean that the reasoning itself is no longer algebra?
  - mathgenius 6y ago
    The ancient Greeks did stuff by example, not using algebra. They said stuff like, "take 22 and multiply by 5..." with actual numbers involved, then at the end they say "and that's how it works in general". So it's up to the reader to generalize from the specific. Algebra works in the opposite direction, where the reader is obliged to specify from the generic. I happen to much prefer the ancient Greek way of doing mathematics, but it seems to have fallen out of fashion.
    - tom_mellior 6y ago
      "The ancient Greeks" are certainly not a homogeneous group. Here's Euclid's proof of the Pythagorean theorem: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI47.html https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI4... There are no concrete quantities in sight. I don't think that they were removed by the translator. Much more plausibly the original reasoning was symbolic in the way indicated here. Though again, possibly using different notation.
lcuff 6y ago
I was disappointed that the article didn't describe some of the motivating forces behind the basics. I think such concreteness would make more sense for a high school student. As an example, the development of geometry was (at least partly) motivated by the need to restore property lines after the annual flooding of the Nile in Egypt. I have no idea if it is part of an "origin" story for probability but wanting to win at dice games is one motivation for understanding probability. One of my heroes, Grant Sanderson, the creator of 3Blue1Brown is very clear on how the concrete needs to precede the abstract in many of these studies, and y = mx + b is pretty abstract if all you've got is those six symbols.
- abhishekjha 6y ago
  You mean 7 symbols.
jchook 6y ago
Gödel's Incompleteness Theorem essentially thwarted Principia Mathematica by proving that mathematics itself is either consistent or complete (read: either useful or definable), but not both. The best definition I've heard calls math "the science of pattern". However, proving itself undefinable seems fundamentally better.
- LudwigNagasena 6y ago
  Just because you can’t assign a truth-value to “this sentence is false” doesn’t mean that language is “undefinable”, the same with math.
- mathgenius 6y ago
  It is such a mystery, how consciousness can perceive pattern.
- ethn 6y ago
  Note this is Whitehead's and Russell's Principia. Although provably a failed project, I've found tremendous value in the texts.
- proc0 6y ago
  IMHO Gödel's Incompleteness is also evidence that what is colloquially referred to as math is just a part of something much larger that we cannot yet grasp.
yholio 6y ago
Platonists don't need to renounce empiricism. Assuming that I am real, if I can think about what mathematics is and arrive at answers, then logic exists and reality is non-uniform, it has some form of structure that permits non-uniformity of answers, allows questions, logic. So the first thought I formulate is an empirical proof that reality has structure. Mathematics is nothing but a description of that structure of reality. We may describe it in human-centric terms and models, but the structure must by necessity pre-exist the mind and is empirically discovered by the action rational tought, not created by the mind. No similar conclusion can be drawn for other non-empirical hypothesis, such as the existence of God. My ability to think does not imply the existence of anything else beside myself, so other empirical proofs must be considered.
- chongli 6y ago
  What’s an empirical proof? I’m not being facetious, I’m genuinely interested because I’ve never heard the term before. To me, a proof is a deductive mathematical argument that takes a claim of the form “if P then Q” and shows how Q is necessarily true when we assume P. I don’t know how to translate from reality into that formalism. All of modern mathematics does not attempt to do so. Instead, it is built up from the axioms of set theory, which take the form of rules for manipulating symbols on paper (or on a computer or even in our minds) without saying anything specific about what those symbols really mean.
- vlovich123 6y ago
  I like that. The way I like to think of it is that mathematics is the language of the Universe. All parts of the universe can be described by this language but not all things described by the language may occur within the universe. While it can be used to describe the reality we experience about our universe, it can also create imagined universes that have only the most tenuous grasp (if any). Like any language it supports fiction and non-fiction applications. There are also rules about how the language evolves that are more strict and defined than human language and the evolution often is more akin to discovery. It’s like the exercise to try to write every single possible English language novel to find Shakespeare. Except instead of stories about human experience we’re trying to identify which random storyline we’re following maps to some fundamental truth about how the universe works. You can sometimes also map fiction to our experience of the world so that it becomes more tangible (eg playing with 4d objects in a 3d video game viewed through a 2d plane). This is no different from reading a fiction book to learn about concepts like compassion even though the story itself is made up. So it’s not either or to me. Both empiricism and Platonism interpretations coexist. Empiricism is what we use to understand what story reality is telling us and how to confirm our comprehension of that story is correct. Platonism is that mathematics itself as a concept is something humans can explore and try to master but exists outside of us and the rules of the language are predefined outside of humanity. Note that just like any language it even has really significant gaps you can’t do anything about (which ironically the language gave us the ability to prove that about itself). That being said the original question this article references indicates a lack of curiosity, which is where my problem with it stems from. And it’s not even like it’s hard to find (although it may require more than a superficial query). “What are practical applications of algebra” is answered in the first few hits. “Who discovered/invented algebra” can be similarly answered. These questions only sound deep if you are actively avoid actually trying to find answers to these questions. There are legitimate philosophical discussions to be had but the original question was not an attempt to have that discussion.
ethn 6y ago
It pains me to repeatedly see debates which were decidedly answered in the 1700s by Kant in the Critique of Pure Reason--with such persuasion to even turn Hume. Transcendental idealism is more or less non-controversial in mainstream philosophy. Especially without even a mention of idealism from the author of the article. Math is a synthetic a priori judgment, a transcendental object. I can only believe that this fashion of thought is from the unwarranted development of the nihilist project which came nearly 100 years later. Bertrand Russell wrote in his historical treatment of philosophy that much of philosophy is forgotten and rediscovered throughout history. I'm skeptical of the argument but less so everyday. >Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist... Roger Penrose states in his book "The Road To Reality" that he is a light Platonist.
- curuinor 6y ago
  If you state that philosophers mostly have a consensus on anything, you'd be brutally wrong. That's not how philosophy is, or can be. Consensuses is not what they do.
  - ethn 6y ago
    Indeed this is precisely the unwarranted nihilism I'm criticizing, where if we were to believe that consensus in philosophy is impossible, so would philosophical progress--and therefore knowledge. This is a libelous claim that all philosophers are engaging in intellectual masturbation explicitly aware that their exercises are to be of no progress. The Pyrrhonists, Diogenes, Descartes and others famously believed in this; but Kant fixes this, making the discipline of philosophy possible by proving the scope of reason.
  - deleted 6y ago
    [deleted]
- btilly 6y ago
  It pains me to see people who are blithely unaware of the philosophical challenges that mathematics faced from repeatedly falling apart in the 1800s and therefore believe that the real challenges were already solved before that happened. If only.
effnorwood 6y ago
A thing that lets airplanes fly
mensetmanusman 6y ago
Poorly written discussion about empiricism. Can you touch concepts? Can you make an object fall upwards against the laws of gravity in your imagination?
jgwil2 6y ago
> If mathematics is just something we dream up from within our own heads, why should it “fit” so well with what we observe in nature? I like to think of mathematics as the formalization of human intuition. Insofar as that intuition has been shaped by our observations of the physical world, we should not be surprised that many of our mathematical models seem to describe aspects of natural phenomena.
ajarmst 6y ago
Of course, the whole platonist/realist/phenomenologist (sometimes characterized as the 'discovered' vs 'invented' argument) is hardly new, nor likely to be resolved (if that were even possible). I certainly recall an enormous amount of cheap on-campus beer being spent exploring the issue when I was an undergrad 4 decades ago. I did find a couple of things quite refreshing in the article, though: (1) Wigner's essay is delightful and I'm glad it's seeing a wider audience. We should be reminded of it every time this conversation comes up, if only for its last paragraph. (2) The reference to the fact that Galileo proved that falling objects accelerate at the same rate, not by some experimental demonstration, but wholly within his mind---and it's a proof that any middle-schooler has no difficulty completely understanding. As a reminder that mathematics isn't symbols and rules, but a way of looking at problems (Eugenia Cheng's book, How to Bake Pi, is an excellent look at this way of thinking and approaching the world.). (3) It managed not to get bogged down in the whole pointless and hypocritical "what is math good for" instrumentality debate, that these sorts of questions tend to devolve into.
sasaf5 6y ago
My favorite definition: Math is the language of the simplest definitions. My second favorite (and controversial): Math is the language of the most useful definitions.
maskedinvader 6y ago
A fascinating topic with profound implications. I have always wondered during day dreaming on the nature of mathematics and what it all means. Highly recommend these 2 videos from this YouTube channel named 'numberphile' discussing this topic. https://www.youtube.com/watch?v=1EGDCh75SpQ https://www.youtube.com/watch?v=1EGDCh75SpQ https://www.youtube.com/watch?v=vA2cdHLKYB8 https://www.youtube.com/watch?v=vA2cdHLKYB8
mjreacher 6y ago
I am reminded of a quote attributed to the German mathematician Leopold Kronecker, "God made the integers, all else is the work of man".
MrXOR 6y ago
Mathematics is beautiful, the queen of science, the language of the nature and the best way to make sense of the world.
InfiniteRand 6y ago
This debate is partly a semantics game of the meanings of "discovered" and "invented" If an idea describes reality does that make it a discovery instead of an invention? What if you think the idea describes reality but it really doesn't. For instance, let's use the sense of inventing as pulling something out of thin air, and discovery as growing an understanding of what is in front of you. Math certainly involves elaboration according to fixed rules which could be seen as discovery. It also involves what-if anxioms that redefine the basis for the mathematical thought (say 4-dimensional space), that could be thought of as invention. This article seems to think of invented and discovered in terms of whether it is not based on reality or whether it is based on reality. This is trickier, I think it is fair to say that the needs of reality do drive certain branches of mathematics, but certain branches are proudly not based on reality. The funny thing is that periodically we realize that the math we thought was firmly not based on reality ends up describing reality better than the math we based on reality. So you have math that was spun out of an idle what-if thought that ends up describing with high accuracy an important physical concept. Is that an invention or a discovery?

133 points by optimalsolver 6y ago

Pick-A-Hill2019 6y ago

An excellent article/submission, one of those huh, never thought to even think about that moments. I especially enjoyed the replies by Eugenia Cheng [1] on points 3 and 7 (from a link in the submitted article). [1] http://eugeniacheng.com/wp-content/uploads/2020/08/gracie-twitter.pdf http://eugeniacheng.com/wp-content/uploads/2020/08/gracie-tw...

loopz 6y ago

We have taxonomies of objects and concepts, but do we have taxonomy of relationships? This might continue to mystify us as long as we try to push the square block into the round hole. It worked so well with the smaller pieces.. Every new discovery started with a question and letting go of some assertions.

mceldeen 6y ago
Category Theory might be a good fit for this. There’s even a category of morphisms (relationships). Perhaps I’m misunderstanding your comment.
aliceryhl 6y ago
Yes. The book Thinking about mathematics by Stewart Shapiro covers it in chapter 10 and calls it Structuralism. https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Structuralism_(philosophy_of_m...

ducaale 6y ago

A video which discusses this subject and compares different schools of thought https://youtu.be/AmySxYHqQCQ https://youtu.be/AmySxYHqQCQ

cloogshicer 6y ago
Came here to post this. It's absolutely wonderful!
3np 6y ago
https://www.youtube.com/watch?v=Kh0Y2hVe_bw https://www.youtube.com/watch?v=Kh0Y2hVe_bw Or, more on-topic from the same: https://vimeo.com/77451201 https://vimeo.com/77451201
- Quekid5 6y ago
  Oh god, I love that Look Around You (What are...) skit so much. It's just so incredibly silly.
ImaCake 6y ago
I particularly like the closing comments about how we teach math. It is something of a theme that we do a bad job of explaining why math should be taught. It took me many years in academic research to realise that not only was math useful to me, but that I liked it. If we could do a better job communicating what math can do - make us rich, predict the future, win at games - maybe it would hold the attention of our students better.
marc_abonce 6y ago
I'm not even epileptic, but that video felt like it would give me a seizure at any moment.

willis936 6y ago

It’s important to note that no model using math has accurately represented nature. Just because simple approximations line up in a beautiful way is not reason to think they are 1:1 the same. Math is not sacred or divine. It’s a tool. Like all tools: its shape matters. Math, as we know it, is invented. It is one set of rules that we like to use to describe things. It is not the only way. I can imagine an intelligent species that has a structured set of logical rules that looks very different from our math but still be able to make useful models.

senthil_rajasek 6y ago
"It’s important to note that no model using math has accurately represented nature" (Edit: I am getting downvoted, so let me expand my answer a bit. Nature is explained using Physics and Physics uses Mathematics for its models of Nature.) This is part of Godel's discovery. Especially, his first Incompleteness Theorem. https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes...
- tsimionescu 6y ago
  The incompleteness theorems don't have anything to do with nature, they are about pure mathematics.
  - senthil_rajasek 6y ago
    Physics explains nature using mathematics.
    - protonfish 6y ago
      And writers explain nature using words.
      - senthil_rajasek 6y ago
        Not as precisely as mathematics can as shown by F = ma or E = mc^2.

spodek 6y ago

One could equally ask what is history, art, sports, etc. The answers mathematicians give seem partly interesting, but also, to me, imply insecurity -- that they have to justify to others that everyone should learn it. Not many other subjects suggest needing justification. Meanwhile, kids teach themselves math in self-directed learning environments https://www.psychologytoday.com/us/blog/freedom-learn/201004/kids-learn-math-easily-when-they-control-their-own-learning https://www.psychologytoday.com/us/blog/freedom-learn/201004.... Up to high school they taught it to me by textbook, mainly teachers who didn't know or care about it. I found beauty in it and loved it despite them. I suspect if we taught it less relying on authority and grades, more engaging and based on curiosity, we'd appreciate it more. Instead of asking what it is, we'd do it. Even people who don't play music don't ask what it is, like it doesn't need to exist. They probably would if we just taught music theory from textbooks but not singing and dancing.

mcphage 6y ago
> One could equally ask what is history, art, sports, etc. Yes, those are all meaningful, interesting questions that people have pondered for a very long time.
TheBigSalad 6y ago
Kids just have no motivation for math. It's not obviously enjoyable like art and music. Teaching what it is and how it helps them in the real world motivates them to learn.
- jng 6y ago
  Some kids do (I did).

aliceryhl 6y ago

Platonism isn't the only way to belive that mathematical truth exists beyond humans. In Platonism one believes that the objects themselves are real in some sense, but I prefer to see it in a different light: mathematics is about considering the relationships between arbitrary objects. The actual objects don't matter - what matter is how they are connected, and the things we prove are true whenever a collection of objects fit together with that relationship.

proc0 6y ago
> mathematics is about considering the relationships between arbitrary objects. The actual objects don't matter - what matter is how they are connected You know this is Category Theory I assume? I guess it does describe math abstractly, but in a way it is also math. Not sure tbh.

wruza 6y ago

Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all? How does one find ethical or religious problems and questions in pure logic? Logic is fundamental and proven to be able to apply in reality. Ethics and religion operate over multiple levels of qualitative transitions and cannot(?) be fundamental. Massimo Pigliucci, a philosopher at the City University of New York, was initially attracted to Platonism—but has since come to see it as problematic. If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?) Because DoJC exists outside of space and time in minds of believers? It just doesn't have a physical counterpart and does barely have any logic to evolve/deduct from it, for us to predict events based on DoJC as if it was applicable to parts of reality. It is understandable how someone could ask "stupid" (not really) questions about maths, but when philosophers start to draw parallels, it is easy to lose track of what they are even talking about. Can someone explain what they meant exactly?

koheripbal 6y ago
There are some core questions in ethics that might be considered so fundamental that they almost don't seem like religious question at all. The Observer Problem is a good example. I experience life from this singular human's perspective. Why this specific one? What mechanism dictates which human my "observer" occupies? Why even a human and not an animal? The word "soul" is a poor descriptor because it is associated with so much more, but the Observer Problem is real - or at least it might be. I heard a theory is that there is another higher order time dimension, and all of our "observers" are actually the same one living in some sequence "simultaneously". ...but that doesnt really solve the dilemma either because even that sequence would need some mysterious ordering mechanism. ...but in all that there are core questions about what is time, what is space, what is energy, what happens to the universe... etc One thing I tell me daughters is that death isn't scary. ...because we exist inna limited space and time. We should be no sadder to not exist in the future than not existing 10m to the left. We cannot be eternal, but if you recognize that time is just a dimension, then our existence is permanant in the history of the universe. Nothing can undo what we did and who we were.

ahelwer 6y ago

I've always liked the idea of math as a type of language. When you learn quantum computing, for example, you go beyond the limits of our informal language (English, etc.) which developed in a purely classical world. This is why pop science articles on quantum phenomena never really make any sense. They try to use metaphors to explain concepts, but there is no existing concept in our informal language to use as a referent. Many quantum phenomena belong to an ontological category that is entirely new to most adults. For example, superposition: you try to explain it as something being in two places at once, but this doesn't make sense nor is it very helpful for understanding how objects in superposition behave. In order to understand quantum concepts you have to learn a new language - that of mathematics. Then it becomes easy! Superposition is just linear combination. Done. Also if you enjoy thinking about this topic I recommend Neal Stephenson's book Anathem, which I just finished. It is a complete slog at the start, with endless boring architectural descriptions. I listened to it in audiobook form which made that section much more pleasant.

dfischer 6y ago
In my understanding, that’s exactly what math is, a language. And various modes of thought have utilized the language of mathematics to communicate ideas that were otherwise not possible or not consistently able to arrive at same conclusions. This is very interesting. What is language? When does language evolve? Is language itself consciousness? Whenever there’s a leap in language is it because something had to be constructed to help assign an abstraction to a coordinate in space where thoughts exist? It appears we’re languages all the way down. Or analogicalmachines? Lol. Lots to unpack here - some with what many would call not nice Consequences. What happens when a minority elite (unsure how else to describe it?) disciplines require a language that the majority lack the finesse to use? Example: math used in physics, or programming in computation. If the languages being used introduce more complexity without the ability to reduce net complexity then it damages the communication of the specie using it. I’ve been thinking about this quite a bit on our trajectory as a species. Complexity in language leads to information silos which leads to various existential risks based on information and miscommunication. Also interesting coercion / authoritarian emergence comes out of those creating a new language to reason about an abstraction which leads to a success that negatively impacts those which do not know the language and if the space to understand said language is sufficiently far enough you have much room for manipulation. Also, who gets to say which language is the right one? What happens when a new language is created that is more advanced and the majority disallow it preventing an evolution? Interesting dynamics.

gumby 6y ago

She asked two important questions, only one of which (the epistemology of mathematics)was addressed in the article. The other one was “why were the Greeks (and contemporaneous Indians) wondering about geometry of a plane and why were the Arabs thinking of algebra” when they had more limited technology than we do?

naringas 6y ago
maybe because they distributed the output of their limited technology better than we do? there's just too much busy-work in modernity
TeMPOraL 6y ago
The latter was covered in the answers by Eugenia Cheng, linked in the article: http://eugeniacheng.com/wp-content/uploads/2020/08/gracie-twitter.pdf http://eugeniacheng.com/wp-content/uploads/2020/08/gracie-tw.... TL;DR: they were curious, they saw some patterns in their daily experiences, and tried to generalize them.

JohnStrangeII 6y ago

This is a rather interesting article. I'm a Platonist. From what we know today, real numbers cannot exist in a finite space, but they seem to exist mathematically. The same can be said about many other mathematical structures, including those that can only be characterized adequately (categorically) in higher-order logic. It's also worth noting that physical and mathematical existence are based on completely different criteria. For non-constructionists mathematical objects exist once they are not demonstrably contradictory (although the absence of contradictions often cannot be proved in an absolute sense). In contrast to this, for physicists an object exists once it can be measured, where measurement is ultimately tied to sensual experience. There are also theoretical entities in physics that cannot be directly measured, but their existence is usually downplayed, they're not supposed to "really" exist but only as theory-dependent entities. In any case,the two "kinds of existence" are very different from each other.

dfischer 6y ago
Quite an astute observation. I’ve been thinking about this quite heavily recently. It has odd occult references too in the sense of “nothing creates something.” That which is measured, exists. No measurement itself, as in of itself as a thing, is a thing that exists. You can measure 3cm but “3” and “cm” don’t exist. Virtual values assigned by consensus. All axioms are agreements. Truth or existence itself is convergent and resembles certainty only in the majority agreeing to it. Imagination allows nothingness (imaginary things) to be measured. Imagine any unit, give it a “rule” and now it can be measured! Pretty interesting bridge between the “imaginary” and the “real.” There’s so much nuance on what existence is, and isn’t, or even not is nor isn’t is depending on if you subscribe to classical logic based on what we currently describe as mostly Aristolian or if you use many-valued logics which are making a come back yet rooted in the Vedas... I find this extremely profound and it’s one of my top focal areas of study right now. It’s also related to my observation of disagreement, chaos, and behavior of the human animal. It seems the zeitgeist of the egregore lacks the ability to adjust perspective and see truth depending on the “rule/roles being played” (all disciplines are games/acting in a way). Tons to unpack.
TheOtherHobbes 6y ago
The perception of numbers exists mathematically. Likewise for more abstract structures. You cannot show me threeness. You can show three things and tell me to generalise, and if you keep generalising you'll end up with a consistent-ish framework of sorts for your observations that can be applied to certain other physical experiences. But fundamentally this is an exploration of the consequences of psychological processes - like perceptual grouping, and inductive relationship inference - not observations of external phenomena. It doesn't seem that way, but there is no external authority you can appeal to which will state definitively that when mathematicians all agree on something their experience of "true" is absolutely and objectively correct, and not a distorted and limited artefact of human cognition.

delhanty 6y ago

>Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist ... That statement surprised me! Google told me that according to this biography by J J O'Connor and E F Robertson [0], Roger Penrose in a debate with Stephen Hawking in 1994 self-described as a realist rather than a Platonist. They quote Penrose as follows: >At the beginning of this debate Stephen said that he thinks that he is a positivist, whereas I am a Platonist. I am happy with him being a positivist, but I think that the crucial point here is, rather, that I am a realist. Also, if one compares this debate with the famous debate of Bohr and Einstein, some seventy years ago, I should think that Stephen plays the role of Bohr, whereas I play Einstein's role! For Einstein argued that there should exist something like a real world, not necessarily represented by a wave function, whereas Bohr stressed that the wave function doesn't describe a "real" microworld but only "knowledge" that is useful for making predictions. [0] https://mathshistory.st-andrews.ac.uk/Biographies/Penrose/ https://mathshistory.st-andrews.ac.uk/Biographies/Penrose/

thinkloop 6y ago

Math is a language that describes relationships between quantities of different things.

jng 6y ago

I like to think about it as this: numbers are a human invention, but quantity is a property of the universe. The laws of physics, including all types of conservation, take care of that. Quantity takes care of ensuring many derived properties (rules) that numbers show and which can be expressed abstractly (complex math). Studying the properties of a complex self-consistent system is interesting, and sometimes, when it reveals an interesting resulting property, it can be traced all the way back to the original natural property of quantity, and applied in a practical way in the world. Other times, it can't, and it remains a curious interesting fact in the realm of the abstract system.

hatch_q 6y ago
I did a thought experiment few years back: Is anything real if we live in a simulation? I came to conclusion that all physical laws, chemistry, the universe would be simulated. And the only "truth" that is the same for host and simulation is math. Imagine that you're creating computer game - a limited simulation. You can change laws of physics, chemistry, life, etc... but math will always hold true. 1 + 1 will always be 2.
- cyberbanjo 6y ago
  Reminds me of Geohotz Breaking the Simulation.

082349872349872 6y ago

The idea of something existing “outside of space and time” makes empiricists nervous This seems like a boring objection: where does the letter 'A', or the ascii 0x41, exist? I'm willing to admit that concrete instances of both exist in space and time (and I might even admit maps forming a galois connection between those and the abstract ideal 'A') but there's no "reference" canonical A, located in time, located in space, giving them meaning.

codegladiator 6y ago
> This seems like a boring objection: where does the letter 'A', or the ascii 0x41, exist? Only in our minds ? A doesn't really exist IMO.

deleted 6y ago

[deleted]

thenoblesunfish 6y ago

One could argue that a defining characteristic of human intelligence is our ability to agree on "unreal" concepts. I think the first time I heard it explicitly was in the book "Sapiens", though I doubt it originated there. In this sense, mathematical concepts are no different than, say Santa Claus. We can agree (mostly) about what it is we're talking about, and talk about it. What sets mathematical "unreal" ideas apart is just how useful they seem to be in describing the universe. They're patterns that fit onto lots and lots of things. For instance, we can all sort of agree on the idea of a (finite) set, and on some basic rules of logic, and with those you can build a lot of mathematics.

dfischer 6y ago

(Commented on someone but they deleted. Will post as parent? All axioms are imaginary. Given that humanities method of logic is imaginary, then consensus is the only difference between certainty and non certainty which is equivalent to truth. Logic is programming. Human’s are programmers and hardware/OS who largely don’t realize they’re running arbitrary truth values based on popular imaginary rules. To exist, is to LARP? Lol.

nendroid 6y ago

Whether math is like an art or whether it is "discovered" is sort of a pointless question to me. To start off, math is simply the art of understanding the consequences and theorems from a set of axioms. Since there are an infinite amount of sets of axioms, there's an infinite amount of math. Whether a set of axioms or the theorems derived from a set of axioms is discovered or created is again a pointless philosophical debate that is not exclusive to mathematics. You can apply this question to any topic. Case in point: There is a finite set of paintings that can be painted on a fixed sized canvas using a fixed set of colored paint. When you create a painting... are you creating a painting or are you "discovering it" from the set of all paintings? Since the question does not pertain exclusively to math; if you think hard you can sort of realize that it's just philosophical speculation derived from intricacies in our language. Is anything truly created? Or is it just "discovered" from the set of all possible things? When we think about this stuff we're not actually thinking about anything profound, rather we are just examining intricacies of language and the definition of the word "discover" and "create." I mean if you think really hard and try to formalize the definition of "create" it can be said that to "create" something is to "discover" a single member from the set of all things. It can also be said to "discover" something is to "create" a new member from the set of all things. People are just getting excited over a language issue. The word "create" was never formalized and forms a sort of fuzzy overlap with "discover." Basically it's one and the same thing sort of like is the glass half full or half empty? The actual reality of this problem is uninteresting and pointless. This paradoxical-like phenomenon doesn't actually exist and arises exclusively because in our language, we have a word called "empty" and we have a word called "full." Consider the black wall. Is the wall "black" or does it lack color? Consider the flying bird. Is the bird flying or is it falling and missing the ground? Is the dark room full of darkness or is light just absent? It's all just language tricks I can "create" or "discover" an infinite amount of garbage here for you to think "profoundly" about without actually realizing it's just a language issue.

tim333 6y ago
Dunno - math seems to underlie the laws of physics and hence existence itself which is not just a language thing.

euske 6y ago

I like the idea of math being a fiction, but the difference between math and novel is that in math you can never ever break the rules once they're set. In contrast, I see that novelists constantly ignore or retrofit the rules to make it interesting. Imagine this was allowed in math! I also think that the fascinating part of programming is that it's sitting in between math and fiction, depending on an application. Sometimes you have to obey the rules very strictly (e.g. writing a parser), but in other times you can be wildly imaginative (e.g. designing UI).

082349872349872 6y ago
Non-euclidean geometry was a retcon.
ahelwer 6y ago
Well, people certainly invent novel proof techniques which remain controversial for over a century. For example, Cantor's diagonal argument, which people still deny the validity of today (although those who do are generally considered cranks).

motohagiography 6y ago

In fewer words than the article, there's a concept called a substrate, kind of like what water in an oocean is to a fish, and while we experience the patterns of it all the time as part of our normal existence, expressing it in language that lets us share our experiences of it with each other is very fiddly. But when we manage to describe things that consistently reflect other peoples experiences, it is like drawing a map. You can experience the territory without it, but it's a totally different game when you have one. So between the critical theory poles of empiricism and Platonism is our ability to perceive and express our experience of our substrate. (What is our substrate? Turtles, all the way down.) Institutions these days never seem to pass up a teachable moment when it comes to reinforcing the precepts of dialectical materialism. e.g. "If something doesn’t have a physical existence, he asks, then what kind of existence could it possibly have? “If one ‘goes Platonic’ with math,” writes Pigliucci, empiricism “goes out the window.” (If the proof of the Pythagorean theorem exists outside of space and time, why not the “golden rule,” or even the divinity of Jesus Christ?)" It's so tedious as to make me doubt their humanity, but I'm glad this girl sparked a popular discussion.

konjin 6y ago

Perhaps a history lesson would benefit her more than one in mathematics. We don't know much about Pythagoras and why he did mathematics. But we do know a bit more about Archimedes and the reasons why he did mathematics: to kill Romans and improve plumbing. Two things that mathematics was found to be extremely good at.

afiori 6y ago

my favourite (less than insightful) answer is that we live in a (hard) magic world and math then is the language of magic in a literary sense, which can then be used to formulate spells in like physic, chemistry, or psychology (we too are magic and can interact with the magic source in our own special way)

pessimizer 6y ago

Math is the series of statements that can be made about things that are true. Math can't tell you that there are three of [anything that you encounter with your sensory organs], but it can tell you that if there are three of those, you can also say... In that way, it tells you about the real world while having no connection to any real world. The problem is that as soon as a math-predicted observation fails, we can blame the mistake on our sensory organs - either formatively (when we assigned a particular observation to a particular mathematical entity (like "three"), or conclusively (when we assigned a mathematical result to a sensation.) If a mathematical result consistently fails to apply to any sensation, it ceases to be math - not by disappearing, but by failing to be consistent; if you can't apply a mathematical process to a sensation in a way that enables you to predict a subsequent sensation, there's no way to determine whether the math makes any consistent statement. Math has to be empirical because it's completely vetted by sensational observations. If a hypothetical god changed some aspect of the underlying physical reality (inclusive of changing some aspect of human sensory organs), the math would change along with it, and things that were math would cease to be math. If you included that god within your mathematics, math becomes the set of all possible maths, which is just the universal set. There's no meaning to a math that says everything about everything. I guess you can say Platonism is true too, since math is a statement about an underlying reality (because it is discarded when it is not), but it's not a meaningful thing to say. Math that turned out to be correct (and continues to turn out to be correct) is Platonic. Math that was incorrect (or in the future becomes incorrect) is not Platonic. The only way you can determine which math is Platonic and which math is not is through empirical observation, and the degree to which you can be sure that particular math continues to be Platonic is through how well your past observations predict your future observations. To answer the tweet, people study math because because being able to predict things gives you an advantage in your work.

yantrams 6y ago

For anyone interested in the topics discussed here, I highly recommend What is Mathematics, Really? [1] by Reuben Hersh [1] https://www.amazon.com/What-Mathematics-Really-Reuben-Hersh/dp/0195130871/ https://www.amazon.com/What-Mathematics-Really-Reuben-Hersh/...

tim333 6y ago
I read that and was a little disappointed. He argues it's basically just a human invention and skirts around the obvious counter arguments that say aliens would discover the same maths if with different language to describe it. He also tries to politicise the debate which I feel is kind of silly.

ArtWomb 6y ago

This is wonderful. I love when memes breed philosophy. We're teaching math wrong. It's as plain as day to me. Or rather we teach the wrong maths to the wrong students. If Marcie can't find a useful application on day one of the "theory" she learns in school, what is the point? It's a 19th century curriculum for a 21st century world. Basic accounting and personal finance would probably serve most students better. As to the epistemological quandary, I always return to Geometry or Visual Mathematics. Consider the infinite ways of calculating the digits of PI. In "Playing Pool with PI", Galperin demonstrated an analog computer that could calculate the digits of PI using only the perfectly elastic collisions of two billiard balls on a pool table. It works for all integral base number systems, and can even be extended to irrational numbers. And recently Google Brain's Adam Brown made an equally stunning observation: Galperin's bouncing billiard balls look identical to the Grover Quantum Search algorithm! Playing Pool with |ψ⟩: from Bouncing Billiards to Quantum Search https://arxiv.org/abs/1912.02207 https://arxiv.org/abs/1912.02207

yeellow 6y ago
> If Marcie can't find a useful application on day one of the "theory" she learns in school, what is the point? It's a 19th century curriculum for a 21st century world. Basic accounting and personal finance would probably serve most students better. Sorry, but following this logic we will end up with a conclusion that almost no school knowledge is "useful". How useful is playing volleyball? Reading poetry? Painting? Studying history? Biology would be reduced to first aid and basic medicine. I have those discussions with my daughters who learn how to do long multiplication. How is it useful if even I who know how to do it would rather use a calculator on my mobile instead? We would reduce math to basic operations on a calculators, probably nothing more would be considered useful by the majority of the population. The same apply to all other subjects.
- unjurios 6y ago
  I think "application" is the wrong focus. The real thing to help students understand is this exact question: what is math? That remains a mystery to most students. It gets conflated with the problem of application, but I think the real problem is most educations don't make clear the idea of abstraction, symbol systems, and logic. When you don't understand something, especially why it might be useful or at least intrinsically interesting, it's hard to get motivated about it.

cskinner 6y ago

The article spends most of its words discussing the reality of math via Platonism vs empiricism while forgetting its own title "What is Math?" If going for a pithy answer, I like tha following quote by G. H. Hardy: A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. I think I originally saw this quote in "A Mathematician's Lament" [0] (also known by the author's name as "Lockhart's Lament"), which has a lot to say about modern math education. [0] (PDF) https://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf https://www.maa.org/sites/default/files/pdf/devlin/Lockharts...

roenxi 6y ago

What is interesting is I'm not sure how many commenters of the video listened to it. The actual points she raised were mostly focused on "why would Pythagoras have used algebra?". And the answer as far as I can see is he probably didn't. Most of the notation we use was introduced in the 1600s-1700s [0]. The concepts we are taught today, and the way we are taught them, would be foreign to Pythagoras. According to wiki, symbolic algebra only emerged in 1200+ with the Arabs (which probably links to why we don't use Greek by default for algebra but we do with geometry). [0]] https://en.wikipedia.org/wiki/History_of_mathematical_notation#Symbolic_stage https://en.wikipedia.org/wiki/History_of_mathematical_notati...

tom_mellior 6y ago
Pythagoras didn't use modern algebraic notation, but I don't think it's correct to conclude from this that what he was doing wasn't algebra. He reasoned about relationships between abstract quantities in ways that we today would write down in algebraic notation. Does a change of notation mean that the reasoning itself is no longer algebra?
- mathgenius 6y ago
  The ancient Greeks did stuff by example, not using algebra. They said stuff like, "take 22 and multiply by 5..." with actual numbers involved, then at the end they say "and that's how it works in general". So it's up to the reader to generalize from the specific. Algebra works in the opposite direction, where the reader is obliged to specify from the generic. I happen to much prefer the ancient Greek way of doing mathematics, but it seems to have fallen out of fashion.
  - tom_mellior 6y ago
    "The ancient Greeks" are certainly not a homogeneous group. Here's Euclid's proof of the Pythagorean theorem: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI47.html https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI4... There are no concrete quantities in sight. I don't think that they were removed by the translator. Much more plausibly the original reasoning was symbolic in the way indicated here. Though again, possibly using different notation.

lcuff 6y ago

I was disappointed that the article didn't describe some of the motivating forces behind the basics. I think such concreteness would make more sense for a high school student. As an example, the development of geometry was (at least partly) motivated by the need to restore property lines after the annual flooding of the Nile in Egypt. I have no idea if it is part of an "origin" story for probability but wanting to win at dice games is one motivation for understanding probability. One of my heroes, Grant Sanderson, the creator of 3Blue1Brown is very clear on how the concrete needs to precede the abstract in many of these studies, and y = mx + b is pretty abstract if all you've got is those six symbols.

abhishekjha 6y ago
You mean 7 symbols.

jchook 6y ago

Gödel's Incompleteness Theorem essentially thwarted Principia Mathematica by proving that mathematics itself is either consistent or complete (read: either useful or definable), but not both. The best definition I've heard calls math "the science of pattern". However, proving itself undefinable seems fundamentally better.

LudwigNagasena 6y ago
Just because you can’t assign a truth-value to “this sentence is false” doesn’t mean that language is “undefinable”, the same with math.
mathgenius 6y ago
It is such a mystery, how consciousness can perceive pattern.
ethn 6y ago
Note this is Whitehead's and Russell's Principia. Although provably a failed project, I've found tremendous value in the texts.
proc0 6y ago
IMHO Gödel's Incompleteness is also evidence that what is colloquially referred to as math is just a part of something much larger that we cannot yet grasp.

yholio 6y ago

Platonists don't need to renounce empiricism. Assuming that I am real, if I can think about what mathematics is and arrive at answers, then logic exists and reality is non-uniform, it has some form of structure that permits non-uniformity of answers, allows questions, logic. So the first thought I formulate is an empirical proof that reality has structure. Mathematics is nothing but a description of that structure of reality. We may describe it in human-centric terms and models, but the structure must by necessity pre-exist the mind and is empirically discovered by the action rational tought, not created by the mind. No similar conclusion can be drawn for other non-empirical hypothesis, such as the existence of God. My ability to think does not imply the existence of anything else beside myself, so other empirical proofs must be considered.

chongli 6y ago
What’s an empirical proof? I’m not being facetious, I’m genuinely interested because I’ve never heard the term before. To me, a proof is a deductive mathematical argument that takes a claim of the form “if P then Q” and shows how Q is necessarily true when we assume P. I don’t know how to translate from reality into that formalism. All of modern mathematics does not attempt to do so. Instead, it is built up from the axioms of set theory, which take the form of rules for manipulating symbols on paper (or on a computer or even in our minds) without saying anything specific about what those symbols really mean.
vlovich123 6y ago
I like that. The way I like to think of it is that mathematics is the language of the Universe. All parts of the universe can be described by this language but not all things described by the language may occur within the universe. While it can be used to describe the reality we experience about our universe, it can also create imagined universes that have only the most tenuous grasp (if any). Like any language it supports fiction and non-fiction applications. There are also rules about how the language evolves that are more strict and defined than human language and the evolution often is more akin to discovery. It’s like the exercise to try to write every single possible English language novel to find Shakespeare. Except instead of stories about human experience we’re trying to identify which random storyline we’re following maps to some fundamental truth about how the universe works. You can sometimes also map fiction to our experience of the world so that it becomes more tangible (eg playing with 4d objects in a 3d video game viewed through a 2d plane). This is no different from reading a fiction book to learn about concepts like compassion even though the story itself is made up. So it’s not either or to me. Both empiricism and Platonism interpretations coexist. Empiricism is what we use to understand what story reality is telling us and how to confirm our comprehension of that story is correct. Platonism is that mathematics itself as a concept is something humans can explore and try to master but exists outside of us and the rules of the language are predefined outside of humanity. Note that just like any language it even has really significant gaps you can’t do anything about (which ironically the language gave us the ability to prove that about itself). That being said the original question this article references indicates a lack of curiosity, which is where my problem with it stems from. And it’s not even like it’s hard to find (although it may require more than a superficial query). “What are practical applications of algebra” is answered in the first few hits. “Who discovered/invented algebra” can be similarly answered. These questions only sound deep if you are actively avoid actually trying to find answers to these questions. There are legitimate philosophical discussions to be had but the original question was not an attempt to have that discussion.

ethn 6y ago

It pains me to repeatedly see debates which were decidedly answered in the 1700s by Kant in the Critique of Pure Reason--with such persuasion to even turn Hume. Transcendental idealism is more or less non-controversial in mainstream philosophy. Especially without even a mention of idealism from the author of the article. Math is a synthetic a priori judgment, a transcendental object. I can only believe that this fashion of thought is from the unwarranted development of the nihilist project which came nearly 100 years later. Bertrand Russell wrote in his historical treatment of philosophy that much of philosophy is forgotten and rediscovered throughout history. I'm skeptical of the argument but less so everyday. >Roger Penrose, the renowned British mathematical physicist, is a staunch Platonist... Roger Penrose states in his book "The Road To Reality" that he is a light Platonist.

curuinor 6y ago
If you state that philosophers mostly have a consensus on anything, you'd be brutally wrong. That's not how philosophy is, or can be. Consensuses is not what they do.
- ethn 6y ago
  Indeed this is precisely the unwarranted nihilism I'm criticizing, where if we were to believe that consensus in philosophy is impossible, so would philosophical progress--and therefore knowledge. This is a libelous claim that all philosophers are engaging in intellectual masturbation explicitly aware that their exercises are to be of no progress. The Pyrrhonists, Diogenes, Descartes and others famously believed in this; but Kant fixes this, making the discipline of philosophy possible by proving the scope of reason.
- deleted 6y ago
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btilly 6y ago
It pains me to see people who are blithely unaware of the philosophical challenges that mathematics faced from repeatedly falling apart in the 1800s and therefore believe that the real challenges were already solved before that happened. If only.

effnorwood 6y ago

A thing that lets airplanes fly

mensetmanusman 6y ago

Poorly written discussion about empiricism. Can you touch concepts? Can you make an object fall upwards against the laws of gravity in your imagination?

jgwil2 6y ago

> If mathematics is just something we dream up from within our own heads, why should it “fit” so well with what we observe in nature? I like to think of mathematics as the formalization of human intuition. Insofar as that intuition has been shaped by our observations of the physical world, we should not be surprised that many of our mathematical models seem to describe aspects of natural phenomena.

ajarmst 6y ago

Of course, the whole platonist/realist/phenomenologist (sometimes characterized as the 'discovered' vs 'invented' argument) is hardly new, nor likely to be resolved (if that were even possible). I certainly recall an enormous amount of cheap on-campus beer being spent exploring the issue when I was an undergrad 4 decades ago. I did find a couple of things quite refreshing in the article, though: (1) Wigner's essay is delightful and I'm glad it's seeing a wider audience. We should be reminded of it every time this conversation comes up, if only for its last paragraph. (2) The reference to the fact that Galileo proved that falling objects accelerate at the same rate, not by some experimental demonstration, but wholly within his mind---and it's a proof that any middle-schooler has no difficulty completely understanding. As a reminder that mathematics isn't symbols and rules, but a way of looking at problems (Eugenia Cheng's book, How to Bake Pi, is an excellent look at this way of thinking and approaching the world.). (3) It managed not to get bogged down in the whole pointless and hypocritical "what is math good for" instrumentality debate, that these sorts of questions tend to devolve into.

sasaf5 6y ago

My favorite definition: Math is the language of the simplest definitions. My second favorite (and controversial): Math is the language of the most useful definitions.

maskedinvader 6y ago

A fascinating topic with profound implications. I have always wondered during day dreaming on the nature of mathematics and what it all means. Highly recommend these 2 videos from this YouTube channel named 'numberphile' discussing this topic. https://www.youtube.com/watch?v=1EGDCh75SpQ https://www.youtube.com/watch?v=1EGDCh75SpQ https://www.youtube.com/watch?v=vA2cdHLKYB8 https://www.youtube.com/watch?v=vA2cdHLKYB8

mjreacher 6y ago

I am reminded of a quote attributed to the German mathematician Leopold Kronecker, "God made the integers, all else is the work of man".

MrXOR 6y ago

Mathematics is beautiful, the queen of science, the language of the nature and the best way to make sense of the world.

InfiniteRand 6y ago

This debate is partly a semantics game of the meanings of "discovered" and "invented" If an idea describes reality does that make it a discovery instead of an invention? What if you think the idea describes reality but it really doesn't. For instance, let's use the sense of inventing as pulling something out of thin air, and discovery as growing an understanding of what is in front of you. Math certainly involves elaboration according to fixed rules which could be seen as discovery. It also involves what-if anxioms that redefine the basis for the mathematical thought (say 4-dimensional space), that could be thought of as invention. This article seems to think of invented and discovered in terms of whether it is not based on reality or whether it is based on reality. This is trickier, I think it is fair to say that the needs of reality do drive certain branches of mathematics, but certain branches are proudly not based on reality. The funny thing is that periodically we realize that the math we thought was firmly not based on reality ends up describing reality better than the math we based on reality. So you have math that was spun out of an idle what-if thought that ends up describing with high accuracy an important physical concept. Is that an invention or a discovery?