38 ms·

Is Math Real?

208 points by hhs 3y ago

booleandilemma 3y ago
I've read other books by this author and felt that, for math books, they were overly political. Is this one any different?
- viscountchocula 3y ago
  Can you elaborate on what you mean by that?
  - keiferski 3y ago
    Not the person you’re asking, but the reviews of previous books make similar complaints: https://www.amazon.com/Joy-Abstraction-Exploration-Category-Theory/dp/1108477224/ref=mp_s_a_1_2?keywords=eugenia+cheng&qid=1695698839&sr=8-2#aw-udpv3-customer-reviews_feature_div https://www.amazon.com/Joy-Abstraction-Exploration-Category-...
    - shepherdjerred 3y ago
      > This is not just a mathematics book, but unfortunately also a polemic of progressive politics. For example, on p.42 there is a helpful definition of “cisgender”, in case you didn’t know. Then, on p.319, the notion of isomorphism — ubiquitous in mathematics, and really quite simple — is explained by considering two categories induced by a partial ordering, one consisting of concepts involving rich, white and male, and another involving rich, white and cisgender. It is then shown that rich white cisgender women have the same status in the second category as rich white males do in the first. Just below this, there is a helpful diagram showing that each of these categories is a 3-fold product of the category consisting of two objects and one arrow, namely (people with structural power —> people who are structurally oppressed). Really, there are many simpler ways to get these ideas across.
      - keiferski 3y ago
        Yeah, I try to avoid the culture war stuff, but this seems pretty unnecessary for a math book.
      - eru 3y ago
        She used the same example in some lecture on youtube, too.
- 4ad 3y ago
  Yeah, I think the mathematical exposition was pretty good, but I couldn't bring myself to finish the book because of the culture war she kept trying to shove in.
  - dagw 3y ago
    Are you referring to this book in particular or her writing in general?
    - 4ad 3y ago
      I am referring to her other book, The Joy of Abstraction.
smugma 3y ago
Is Math Real? Not when we are talking about how Math can be Complex.
- deleted 3y ago
  [deleted]
- quickthrower2 3y ago
  It can also be Rational. They hate to admit it, but it is all Imaginary.
moritzwarhier 3y ago
This introduction is a very pleasant read and adequately dense and easy at the same time.
tabtab 3y ago
Math is merely a mutual agreement. First an agreement on root principles, and then an agreement on notation for these root principles. The rest is built on top of those. Nobody is forced to accept the premises. (Unless you don't want an F in math class.)
- dekhn 3y ago
  One imagines two intelligent alien races would be able to communicate that pi is the ratio of the circumference to the diameter of a circle without coming to any real agreements over postulates or axioms. But I suppose that's natural philosophy, not math. See https://computingbiology.github.io/docs/hamming1998.pdf https://computingbiology.github.io/docs/hamming1998.pdf "Mathematics on a Distant Planet" for more thinking. Many physicists think that math is instrinsic and absolute (objective) in the universe, not "merely a mutual agreement"
  - proamdev123 3y ago
    > One imagines two intelligent alien races would be able to communicate that pi is the ratio of the circumference to the diameter of a circle without coming to any real agreements over postulates or axioms. Except circles don’t exist in the real world. They are an abstraction that mankind created that are similar enough to many natural objects as to be a useful abstraction of them. In order for two alien races to communicate about pi, they’d have to first agree on the definition of a circle as a set of points equidistant from a central point. That’s one of the axioms they’d have to agree on before they could even get to pi.
    - jaza 3y ago
      Assuming that the two alien races are indeed "intelligent" (approximately relative to humans), and that they possess sensory perceptors approximately similar to those of humans, and that they inhabit the same observable universe as humans do - a universe that contains a great many physical (imperfect) circles - surely they should be expected to have conceived of the same abstraction of a (perfect) circle? And therefore to have gotten to pi? Then again, maybe a circle (and all other geometric abstractions) are too tied specifically to visual perception. Maybe, for example, intelligent aliens who only have auditory perception, would have no concept of a circle but would have gotten to the Doppler effect. Or intelligent aliens who only have tactile perception, would have no concept of a circle but would have gotten to the laws of thermodynamics.
A_D_E_P_T 3y ago
Mathematics is, in part, the study of universal traits: (1) Ratios between things in the world. (2) Logical relations between things in the world. (3) Absolute distinctions between things in the world, in a nominative sense, which is to say in the sense that numbers can be assigned to things or elements of things. Mathematics that relates to one of the three use-cases above is absolutely real, hence its unreasonable effectiveness in the natural sciences. (See Wigner: https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf ) When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal. A construct or game played with logic that has no prior or intrinsic relevance to the material world.
- bawolff 3y ago
  So when an application is found to a previously pure theory area (e.g. number theory and RSA) does math becone real? Is realness not an instrinsic property but just a judgement on how useful something is? Is being real a real property of something or just a construct?
  - beezlebroxxxxxx 3y ago
    Math is in language. In a similar way that we can have fiction and non-fiction, with different uses of language, you can have 'pure' and 'applied' uses of math. Using 'real' as the goal kind of misses the point. It's not imaginative, if that's the antithesis of real in this whole conceptual discussion. But, like fiction to non-fiction, it's still a public phenomenon that "touches" the world in language as a shared, communal, element in a form of life. Math is remarkably useful as a representation of the world and for us solving problems, but you can also give directions to a location to someone with just hand signals and barking like a dog. In a way, math is not special. But, the degree of accuracy that we've developed in math, however, is what makes it remarkable.
    - bawolff 3y ago
      > Using 'real' as the goal kind of misses the point It misses the point of the question "is math real"? Tbf, i am actually sympathetic to the position that "realness" is not really relavent or a well defined term when it comes to what is basically a descriptive language of patterns and relations. But "is it real" is the question we started the thread on.
kdottt 3y ago
“The question “is math real?” is answered in the epilog of this book. Cheng tells us that math is real because it is an idea and ideas are real.” Alright great lol
- dasil003 3y ago
  What is “real”? Well, it’s just an idea… really
  - dekhn 3y ago
    There is, almost certainly, an objective universe that exists outside of human minds, and it also seems likely that all of mathematics can be determined through objective mechanisms.
- chongli 3y ago
  This is an idea that goes back to Pythagoras and Plato. Think of a triangle. Now draw that triangle on paper. If you look closely enough, you'll see "imperfections" in the triangle you just drew. Now ask yourself: "how do I know this thing I just drew is imperfect? Where did the idea of a perfect triangle come from?" Plato would say the perfect triangle comes from the realm of "forms". This mystical place which is "more real" than "reality" because everything there is perfect and everything here is just a flawed approximation. Plato also said that this is the place where our souls go when we die, and we engage in "congress" with the forms and then return to earth, reincarnated. When we learn things, we aren't learning something new but actually recalling memories of the forms. This is why everyone knows what a perfect triangle is but no one has ever seen one in the physical world.
  - dr_dshiv 3y ago
    Because the Platonic soul (psuche) is different from consciousness. Actually, Philolaus (first Pythagorean to write a book) said that conscious feelings come from the combination of the mathematical soul with the body. So, Pythagorean soul can be viewed as the set of logical or conceptual forms— which constitute a person, yet can also be passed on from person to person. That’s a very different kind of reincarnation…!
RagnarD 3y ago
You might as well ask "Are ideas real?" in the broader philosophic sense. Yes, they (with math as a subset) are real existents within consciousness. The question of mathematical ideas' connection to reality is epistemological. "Real reality" pertains to metaphysics - the externally perceivable stuff of reality that exists independent of our minds (as in, eternally prior to, and subsequent to, the existence of any humans or other beings with conceptual consciousness.) Math is an epistemological tool to abstract the causality of existence into a simplified structure amenable to consideration and manipulation by human consciousness. A quadratic equation can be used to approximate the coordinates of the trajectory of a thrown object in a gravity field. I think the really interesting question is why that particular equation reflects that trajectory. Arithmetic operators - multiplication and addition in that case - in a particular order, are approximating the causal operations of existence that are actually at work. I think of this as the philosophy of mathematics and much more should be done to investigate it.
- coxmichael 3y ago
  > Math is an epistemological tool to abstract the causality of existence into a simplified structure The mathematics to accurately predict or relay reality is still complex enough that it’s often beyond us. You’re right in that it’s a tool to understand, but if we’re using simplified math for simplified reality, is it really epistemological? As you suggest, math isn’t outside the boundary of philosophic investigation. It never was in the past, and I don’t think better approximations change that calculation.
  - balder1991 3y ago
    You made me remember this article you might have read before: https://www.wired.com/story/our-machines-now-have-knowledge-well-never-understand/ https://www.wired.com/story/our-machines-now-have-knowledge-...
    - coxmichael 3y ago
      Thanks, I’ll give it a read
tracerbulletx 3y ago
Brains model patterns in signals, math could be described as the brain's language for expressing higher order manipulations on all those models with communicable symbols. So it's kind of both real and invented, because brains invent, and brains are also just real objects and any models they form came from somewhere.
- User23 3y ago
  The statement that brains model patterns rests on the supposition that patterns to be modeled exist.
  - tracerbulletx 3y ago
    If there isn't then I guess we're all just talking to ourselves anyways so it doesn't really matter.
    - glennonymous 3y ago
      Am I the only person who hates it when people say “anyways” when they mean “anyway”?
      - bloak 3y ago
        Perhaps it's acceptable in moderation, like "ain't". It's needed for a rhyme in a well-known song by Billy Joel, for what it's worth. On the other hand, there is a different song by Billy Joel in which "anyway" is used for a rhyme. No, I'm not a Billy Joel expert or fanatic. I just happened to notice, all right?
      - pnut 3y ago
        That stray character doesn't change the meaning of the sentence, so yeah, stylistic complaints are subjective and beside the point. We're not in school.
  - reuben364 3y ago
    The notion of existence is doing a lot of the leg work. A epistemological solipsist can say things "exist" when they have a model of their subjective experience in which things can exist. It gets real funky because your interpretation of qualia results in a model that includes the brain and thus a model of how you are interpreting things as a model in your model.
agumonkey 3y ago
is math memory ?
autokad 3y ago
> Oftentimes, math is taught as a set of rules and processes divined from an authority in which learners must memorize the rules and strictly follow the processes in order to satisfactorily find the one and only correct answer I was surprised later in life when the majority of people I talked to felt Math was this way (and a good thing). To them, math is a set of rules you learn to follow to the T and use them on other problems. For me, everything a Math teacher conveyed was more of a recommendation, a suggested tool that I could incorporate into my tool box. The methods they employed to solve problems were a matter of preference to me, rather than rigid rules. This way of thinking has always had some pros and cons. I never solved a problem the way a 'grader' was expecting. Some teachers loved the creativity and efficiency, other TA's just marked as zero. It made applying what I learned to other things, but on exams I would always be stressed with time because I would spend time on which way I was going to answer the problem. That being said, I am very good at Math. scored well in HS/college, 165/170 Math GRE score, in a field where Math is important (Data Scientist).
dingosity 3y ago
"Is Math Real?" Define real.
- Cerium 3y ago
  Not having an imaginary component.
  - galangalalgol 3y ago
    I read the comments here just to verify I would find this joke.
    - sacnoradhq 3y ago
      ◻
      - quickthrower2 3y ago
        QED, for the oldies
  - quickthrower2 3y ago
    And in addition a real number can encode an infinite amount of information, and as such real numbers are a construct that doesn't exist in the physical universe. Making them, unreal.
jacurtis 3y ago
This book is on my reading list. I listened to an interview of the author who said that their primary motivation of the book was less about teaching math, but teaching the value of looking at the world abstractly to determine patterns and constructs to explain our shared experience. Is math real or not? It doesn't matter she posits, it works and continues to work and we can learn from its existence that almost everything can be explained and is "predictable" given enough inputs.
itvision 3y ago
Really not the best title for this book.
conformist 3y ago
I've encountered this argument before, but also applied to other subject such as eg Latin: > 3) math has an indirect usefulness which is a way of thinking that is transferable to a myriad of disciplines and solutions to problems in everyday life. And it is this third reason that makes math relevant for most people. Are there actually any good studies that show this in a counterfactual setting? Like, do we actually know that spending time teaching maths (or, less plausibly, Latin) helps students aquire these abstract skills more than other subjects? Is this "transferability" of meta-skills a testable outcome?
- JackFr 3y ago
  I’m skeptical of (3) as well, and yet I encourage students to study math through calculus, and to study Latin. My feeling is that the utility of higher math, Latin, history and literature comes every minute of every day, as you experienced life as someone who has familiarity those things and your life will be richer and fuller. This is decidedly not testable. And yet I still believe it.
  - fiforpg 3y ago
    That (3) is a very interesting issue. My suspicion is, it basically says "knowing logics is good for your well-being". The skill of logical inference — even at the level of very basic syllogisms — is both very much underappreciated and underdeveloped in the American college population, at least from my personal experience. As good citizens, we all collectively should grab a couple of Martin Gardner's or Lewis Carroll's books off the shelf and give them a good read. I predict it will do much good... and if I'm wrong, it certainly won't do any harm!
  - m463 3y ago
    There are plenty of ways to use this stuff in everyday life. I took logic in college, and although the "logic as english statements" stuff was sort of confusing, the symbolic stuff like A&B = !A|!B stuff has helped with computers all my life. It was only much later in life that I ran back into logic as english statements in a way that made sense as practical. I read a book where they took apart the statement: If you loved me, you would take me to the movies. Most people in relationships will respond to this with: Well, I just took you to the movies last week! Why do you want to go again tonight, we had other plans! etc... But the book explained that with "If X, then Y" it was futile to address Y. You must address X: Wait, do you think I don't love you? Of COURSE I love you! ...just hard to unpack this in the middle of an emotional situation unless you've studied the logic :)
deleted 3y ago
[deleted]
ants_everywhere 3y ago
In my opinion, a more interesting question is "is anything other than math real?" Is there anything that physically exists that is non-mathematical?
- the_af 3y ago
  Well, what do you mean when you say something that "exists physically" is "mathematical"? Depending on your answer, the answer to your question could very well be "everything".
  - ants_everywhere 3y ago
    For example, we could ask whether anything that we think of as "existing" would be missed by simulating the equations of physics with the empirical starting conditions of our universe. There are real limits in what humans know about the universe. But it's possible in theory that everything that is "real" follows from some differential equations.
    - the_af 3y ago
      In theory, maybe. I don't think our models of the universe are anything other than imperfect but useful approximations though. Could your question be rephrased as "take the real world, with its things we don't have an accurate model of (which is most of them). Would we notice if we replaced this universe by a tidier one, perfectly described by a math model?". It's an interesting, scifi question alright. I don't know if it has an answer.
- cool_dude85 3y ago
  What would it mean in this context to be "non-mathemtical"?
  - ants_everywhere 3y ago
    I mean that it can't be completely described by mathematics
jjgreen 3y ago
... the typical working mathematician is a Platonist on weekdays and a formalist on Sundays Philip Davis, Reuben Hersh, "The Mathematical Experience"
- User23 3y ago
  Meanwhile the actual working mathematicians, better known as engineers, still use geometry on weekdays.
  - eru 3y ago
    Engineers are their own special species. They aren't just 'working mathematicians'.
ftxbro 3y ago
some of it is integer
PrimeMcFly 3y ago
Yes. Next question.
fiforpg 3y ago
Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among working mathematicians and physicists[1,2], and it is rather clear that the question of reality of a language is ill-formed. A rich enough language should be suitable for expressing both real and abstract statements. It is also interesting to think about mathematics from the point of view of programming language paradigms: is it functional (I think mostly), what is its type system, etc. [1]: Galileo says that, roughly, the book of nature is written in the language of mathematics https://en.m.wikipedia.org/wiki/The_Assayer https://en.m.wikipedia.org/wiki/The_Assayer [2]: A classical essay by Wigner on the "unreasonable effectiveness of mathematics" also brings up the language analogy. https://web.archive.org/web/20210212111540/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html https://web.archive.org/web/20210212111540/http://www.dartmo...
- worldsayshi 3y ago
  So language isn't mathematics? Like physics "is" mathematics? Are both language and math a third thing? Like maybe automata?
  - winwang 3y ago
    considering that formal languages are part of math, there's little reason to believe natural language isn't. ft. LLMs.
  - dexwiz 3y ago
    Mathematics is the language best suited to communicate Physics. > The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed. Translates to > F=ma
    - candiddevmike 3y ago
      Now translate that back to the original English without any context. I don't think your strawman is as succinct as you think it is.
sravic01 3y ago
Unfortunately, it takes extraordinary effort by the math teachers to make Mathematics interesting to students. Even if some of the teachers are willing to put the effort in teaching this wonderful subject in a way that makes sense to students, it's not possible to turn all mathematics teachers into teachers who understand how mathematics MUST be taught. I hope all these books, YouTube videos, and websites can help in making students curious about mathematics and explore further on their own. It is hopeless to even expect the school systems in US and a lot of other countries, to change the way their school systems work - teachers are not given enough time to spend on the topics in mathematics. It takes lot of extra effort by the students.
- User23 3y ago
  The actual reality is most students don’t have the cognitive horsepower for much past long division. I’ve seen adults struggle with basic algebra: “but what is X?” And so on. And that’s fine. Billions of people are going to live satisfying and fulfilling lives without ever knowing the chain rule. Edit: I thought it was pretty advertent.
  - burnished 3y ago
    I think you inadvertently highlighted what I suspect is the real issue; a ruthless focus on the path to calculus to the detriment of other useful maths.
    - otabdeveloper4 3y ago
      Other maths are more difficult to grasp than calculus. Calculus at least has physical applications and can be visualized.
      - burnished 3y ago
        I think I disagree! Calculus is great and all (truly), but logic, boolean algebra, probability, statistics, even linear algebra all have really interesting and useful insights and are very approachable. I'm sure there are other maths I've never encountered that can also be enjoyed with minimal pre-requisite knowledge, though that might be more in the 'abstract and difficult' territory. The bits you need from calculus to approach those other subjects is also very minimal and approachable relative to the content of a calculus pre-req (where applicable).
GuB-42 3y ago
Math is rational, therefore math is real. Because ℚ ⊂ ℝ of course, but it is also not that far off from the conclusion of the book in that "math is real because it is an idea and ideas are real". In a non-mathematical sense, rational means "based on reason", and reason is the power of the mind.
- tomrod 3y ago
  I love your response. But can math be both rational AND irrational?
  - elliotec 3y ago
    Yes. But does irrationality imply non-reality?
    - jacurtis 3y ago
      I work with several individuals who are consistently irrational and frustratingly real.
    - slowmovintarget 3y ago
      Math can also be imaginary, but this also seems to be a feature of the universe.
    - tomrod 3y ago
      Not for a numberphile.
  - cgriswald 3y ago
    In some cases he's rational, irrational, but not real: https://en.wikipedia.org/wiki/Math_fab_Mathonwy https://en.wikipedia.org/wiki/Math_fab_Mathonwy
  - jaza 3y ago
    Only if it's Schrödinger's math
beiller 3y ago
I think I heard that all numbers in the real world are irrational. So that means most of math is not real, except of course the irrationals like pi :)
- adastra22 3y ago
  That seems like a very weird statement to make. Many numbers in the real world are integral (two objects, one electron, etc.). Thanks to quantum physics, most measurements are integral too.
  - beiller 3y ago
    Explained better above. There are more irrational numbers, almost guaranteeing any number come across in nature is irrational. Interesting thought since I think one thing that makes irrational numbers is there is no function for them. So it's kind of a cheeky way to say no math formulas can ever describe the real world since all the numbers are irrational.
    - dagw 3y ago
      There are more irrational numbers, almost guaranteeing any number come across in nature is irrational There is nothing that says that the distribution between rational and irrational numbers that show up in nature is the same as the distribution in our construction of the real numbers.
    - adastra22 3y ago
      You’re missing the point: physics has shown the world to be discretized. Almost every number you go out and measure IS ACTUALLY AN INTEGER. In that sense the real number system doesn’t exist. It’s super useful, yes, but it’s an abstraction away from reality.
      - 2snakes 3y ago
        I thought that "infinities of infinities" ala Cantor exist though. I agree with you that reality being quantized means everything in physical reality can be normalized to integers.
EpiMath 3y ago
This book is not really addressing the more common "is math real" question of it being empirical or invented. For an interesting take on that question, see the 1st section of the 2nd part of Daniel Shanks' Solved and Unsolved Problems in Number Theory. He makes some interesting points about the old Pythagorean views
- dr_dshiv 3y ago
  Pythagorean ideas—- well, it has been some of the most enriching philosophy I’ve ever encountered. It’s a rabbit hole, for sure.
  - eru 3y ago
    It's interesting that mathematicians, when asked philosophically, might have all kinds of interesting and nuanced ideas about this topic. But when you let them get back to their mathematics, they behave as if they believed deep down in their heart that mathematics exists independently of the observer. (It's a working attitude that works well in practice. Just like a heliocentric world view works well enough for most celestial navigation you can do without computers.)
    - quickthrower2 3y ago
      I am waiting for the universe to do something non-mathematical. There is no reason to believe it can't :-)
    - mbork_pl 3y ago
      Sorry to break it to you, but most mathematicians don't give a s*t. Even worse, a large percentage of mathematicians (not sure if "most", but I'm afraid yes) do not usually have "interesting and nuanced ideas" (nor opinions) on anything.
- hhs 3y ago
  > This book is not really addressing the more common "is math real" question of it being empirical or invented. Please note, this is mentioned at the beginning of the review: "I settled in to read the book “Is Math Real?” expecting to become embroiled in the age-old controversary of whether math is invented or math is discovered. Instead, I found myself confronted with two viewpoints of mathematics: one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively. The premise of “Is Math Real?” is that people have different emotions about math. Some love the math and have little difficulty determining the correct answer to a problem while others loathe and dislike the math and have a difficult time ascertaining the correct response. Many times, a student is humbled or chastised for asking ‘a stupid question’. Author Cheng states that there are no stupid questions. In fact, the most profound concepts in mathematics are learned from asking the simplest of questions.”
olliej 3y ago
It is if you use pascal :D
- fiforpg 3y ago
  Do you mean Pascal the person? :P
  - olliej 3y ago
    I was making a terrible joke about the old “real” type for floating point in pascal. Or possibly PASCAL :D
captaincrowbar 3y ago
Math is integer, unless declared real.
- imchillyb 3y ago
  Definitely found the Fortran guy.
  - sacnoradhq 3y ago
    Double down on C.
- sacnoradhq 3y ago
  That's irrational and imaginary. Oh, and the cardinals will win the series.
- deleted 3y ago
  [deleted]
readyplayernull 3y ago
Well, think the other way around: if reality didn't keep the relationship between ratios as a coherent structure, it would be a soup of pure timeless randomness, producing lots of short lived Boltzmann's brains... so, we wouldn't notice.
- didibus 3y ago
  I remember reading of a theory that kind of said that in any large enough random set, over long periods of time, you can identify clusters which will have consistent and predictable behaviors, but only within the bounds of the cluster and for a set period of time, eventually drifting to something else. I thought there was mathematical backing of this observation. Which means it could all be just random, but we happen to exist at such a small scale within it all, and for such a short period, that we're able to find laws applicable to our pocket and our time period that appear to be true and can predict behavior. Thought it was interesting.
  - ashbee 3y ago
    Definitely interesting. I dislike thinking about such things though. I've been stuck on this one for a while: Descartes posited, "I think, therefore I am" as a reassurance of our existence in a given moment. While this anchors our immediate awareness in some semblance of reality, it doesn't necessarily offer solace regarding the past or the future. Imagine, if you will, an expansive universe characterized by its sheer randomness and chaos, existing for eons. If, in such a universe, a consciousness can spontaneously emerge and dissolve—much like a fleeting set of physics—how can we be certain of its duration? Does it persist for a lifetime, or does it flicker for just a moment? Our memories might suggest a continuous existence, but what if they are merely ripples in this vast sea of randomness? It's an unsettling thought: could it be that we exist only in the fleeting now, borne from the precise alignment of countless variables just to perceive this very instant? And that perhaps, a moment ago or a moment hence, we simply weren't and won't be? While such thoughts might be disconcerting, their implications are, in a way, irrelevant. Even if I could unequivocally prove this transient nature of existence, our ephemeral consciousness would barely have time to grasp it. Our understanding would be obliterated, only to be replaced by another random state. And yet, if this fleeting moment is all we have, I'm honoured that you spent it reading this randomly generated comment. Not that you had much choice in the matter. I think it was Shakespeare who said: it was the best of times, it was the blurst of times.
drdec 3y ago
A better question is - is math an inherent part of the universe or is it a uniquely human construct?
- otabdeveloper4 3y ago
  It can be both.
- mensetmanusman 3y ago
  The universe has a mind (human) and its current opinion is that math is inherent and discovered.
- rramadass 3y ago
  Mathematics is an inherent part of the Universe. What we Discover/Invent are Models/Notation for identifying and using them. The difference today is that we have built and extended the Abstractions/Relationships to such an extent that people feel it is not "Real" which is of course not the case. The best argument for this are the various structures across the World (eg. Pyramids in Egypt/Central-Latin America, Temples/Structures in India, Aqueducts from ancient Rome, Great wall from ancient China etc.) spanning thousands of years which could not have been built without a knowledge of Mathematics as we define it today. Their approach and models/notation may have been different but the essence of the Mathematical Abstraction is the same.
  - meroes 3y ago
    There’s so much more math though. More than will ever be seen or used in structures, thought about, etc even in principle. E.g. if string theory isn’t real the math still is, so where is it? Part of the universe is pretty unsatisfying.
    - rramadass 3y ago
      >There’s so much more math though. True, that is why I used the words "Discover/Invent". Also i used physical structures as an example since they are the most visible and unarguable evidence of "Real" Mathematics from the earliest times. You can invent abstractions to model concrete things(eg. all that is needed to model a skyscraper) or to model still further abstractions in a chain (eg. vector spaces for multidimensional/functional/etc. spaces). We only realize that it is "Real" when it is "Applied" in the concrete World (eg. number theory in cryptography).
drbojingle 3y ago
Is value real? Does it matter?
mensetmanusman 3y ago
Is it knowable if anything is real? What does knowable even mean?
- russdill 3y ago
  If math is real, nothing is real.
- deleted 3y ago
  [deleted]
- fifticon 3y ago
  in particular, what does 'mean' actually mean?
corethree 3y ago
Here's the irony. In science and therefore reality as we know it nothing can be proven to be true. Things can only be falsified in science. This occurs because if we make 10 million observations that verify a hypothesis we still haven't proven anything to be true because there always exists the possibility that a subsequent observation falsifies the entire hypothesis. While we can't prove anything in reality is real, we can prove things to be true in mathematics. Proof is the domain of math and logic not science. Therein lies the irony. We don't know if anything is real in science and therefore reality as we know it but we can verifiably know whether things are real in the universe of math. It really puts the question in perspective: what does it even mean to be real?
- bandie91 3y ago
  > what does it even mean to be real? exactly the question which has to be answered before answer "is X real?"
- Warwolt 3y ago
  But this just isn't the case, for your claim to be true we would have to be constrained to only do _inductive_ reasoning. When we do _deductive_ reasoning, and we feel as though the reasoning has been done correctly, then we know something _assuredly_. Mathematics and empirical science are very different in this respect, that much of mathematics is purely deductive.
  - corethree 3y ago
    Inductive reasoning is by nature probabilistic so it can't prove anything. See: https://en.m.wikipedia.org/wiki/Inductive_reasoning https://en.m.wikipedia.org/wiki/Inductive_reasoning The first paragraph mentions this. As science is statistical in nature inductive reasoning by being probabilistic suffers from the same problem.
    - Warwolt 3y ago
      Your claim was that we can only know about reality through inductive reasoning, and this is the part I'm disputing, not that science itself is not inductive. Mathematical reasoning is deductive, and mathematics is part of reality.
keiferski 3y ago
A good starting point for questions like this is the SEP article: https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/
photon_lines 3y ago
If anyone doubts whether math is 'real' or not - maybe watching this clip of these 4 men might clarify it (pay attention to the language they use to discuss the architecture of the universe): https://youtu.be/AmUI2qf9uyo?si=6B6qA249jZqCVV5j https://youtu.be/AmUI2qf9uyo?si=6B6qA249jZqCVV5j
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redder23 3y ago
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slmjkdbtl 3y ago
How does the word "real" serve practical purpose, besides for the fun of logic / language game? It's a tool that we use and we found useful.
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WheelsAtLarge 3y ago
It's a tool. So it's as real as any other tool we have. It helps us bring about our thoughts in to the real world. We can't touch it, in the same way we can't touch language, but it's useful and that makes it real.
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SomethingNew2 3y ago
Based on the Amazon reviews, this doesn’t look like a very good book. “A Lot of Non-math Digressions” “Way too wordy and discursive. Boring format.” “It's hard to imagine that someone would write a book ostensibly about math that has so much personal, political, social, and just plain silly off-subject commentary. This book is so interspersed with that kind of commentary that it is difficult to focus on the math.” “The book has some helpful insights regarding teaching elementary students basic mathematical operations. However, it is filled with so many abrupt political digressions, often incoherent, and ironically, occasionally illogical, that the entire work is barely readable.”
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bowsamic 3y ago
I can already tell that a vast majority of the replies here are going to respond to the title and not the post
- tempaway33751 3y ago
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- crabmusket 3y ago
  > The question “is math real?” is answered in the epilog of this book. Cheng tells us that math is real because it is an idea and ideas are real. Solved!
  - bowsamic 3y ago
    This is not an alternative to people reading the article
    - crabmusket 3y ago
      Agreed. I just thought this little paragraph at the end of the article, which does directly address the title of the article, was amusing.
  - Lockal 3y ago
    If ideas are real, do you have any idea about entities that are not real? I mean, what is his exact definition of "unreal" things? Something that can not be formally described?
    - crabmusket 3y ago
      I'll let Parmenides take that one > It needs must be that what can be spoken and thought is; for it is possible for it to be, and it is not possible for what is nothing to be.
nologic01 3y ago
> one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively I havent read the book but this dichotomy is not an intrinsic feature of how people pursued mathematics historically. The "stiffness" and excess focus on rigor and accuracy developed gradually over the 19th century because people were being loose canons - primarily around calculus.
- leetcodesucks 3y ago
  Its still not stiff enough. Mathematicians really ought to settle on a sound and coordinated syntax
  - eru 3y ago
    Syntax is about the least of people's worries when they worry about 'mathematical rigor'. Math is made for people to read and write, and they have different 'domain specific languages' for different parts of math. Now what would be useful is a tool, perhaps something like a large language model, to automatically translate from the notation used in one area of math into another. That can't be a fully mechanical procedure (hence the need for something flexible like an LLM), because it's part and parcel of human mathematics to abuse notation here and there in the name of ergonomics. My personal pet peeve is defining a function like f(x) := x + 3, and then treating f(x) as the name of the function, instead of just f. But really, it's just a harmless abuse of notation when done by some humans to other humans.
    - rudy6912 3y ago
      I do not think mixing up 'f(x)' and 'f' is harmless, given mathematics is all about clarity. In any quality text, 'f(x)' denotes the value of 'f' at 'x', and 'f' denotes the function 'f' itself. Also, speaking of notation, I wonder why you used ':=' instead of '=' to define 'f'. There is no computation going on, right?
      - 3y ago
acec 3y ago
No if you use sqrt(-1)
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cdevries 3y ago
Everything is real by definition
DarkNova6 3y ago
No. Math is Complex.
- auggierose 3y ago
  No. Math is Surreal.
  - anilakar 3y ago
    Elephant to the melting clockth power equals what?
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jongjong 3y ago
My view is that Math is real but the way it's represented is arbitrary. I'm sure that math could have looked very different than it does today and it would have worked better for a different kind of people. I find that some mathematical abstractions and ways of describing certain ideas are over-complicated and almost seem to be intended to confuse and obfuscate rather than to clearly explain. I feel similarly about certain fields of science; the ideas are really very simple but whenever you hear someone explaining them, the explanation seems to be full of gaps, rely on the audience to make certain assumptions, use custom terminology or terminology repurposed from different fields which can create confusion. I often felt that being good at math requires having a natural talent for making the right assumptions to fill in any gaps or ambiguity in explanations. I feel like the language of math does not do justice to the complex, intricate ideas it tries to convey. On the other hand, being good at programming requires the opposite; it's about being able to resist making assumptions.
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- AvImd 3y ago
  That's really interesting, can you give some examples?
  - jongjong 3y ago
    I feel like almost all the descriptions of advanced (university level) math concepts are confusing when you first hear them. For example, the definition of a Taylor series according to Wikipedia: "In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point." This kind of description really throws me off. Especially "terms that are expressed in terms of the function's derivatives at a single point." first of all, superficially, the different meanings of the words 'term' and 'terms' throws me off and then even when I get over that, it's not clear what is meant 'in terms of' in what terms? What kind of relationship are we talking about here? I need to keep reading a lot more to fill that gap... In the meantime, I'm in a state of confusion and will need to re-read that sentence later to make sense of it once I have more info. I'm not sure how to solve that problem to explain it more clearly but it feels like definitions should gradually build up without gaps. Is it even solvable? I would prefer not having read that definition at all TBH as it only serves to confuse me, it has way too many possible interpretations. I prefer to jump straight to the formula.
ivanche 3y ago
I don't know if it's real but it's definitely complex.
- ShamelessC 3y ago
  Get out.
- tannhaeuser 3y ago
  OTOH it can also be rational and natural.
  - thsksbd 3y ago
    And, amongst the reals, the irrational infinitely outnumber the wholesome
  - ivanche 3y ago
    OTOOH sometimes it's irrational and transcedental so there's that.
- otikik 3y ago
  Came here looking for this comment, not disappointed.
kiriberty 3y ago
If math is not real, it should be complex then.

208 points by hhs 3y ago

I've read other books by this author and felt that, for math books, they were overly political. Is this one any different?

viscountchocula 3y ago
Can you elaborate on what you mean by that?
- keiferski 3y ago
  Not the person you’re asking, but the reviews of previous books make similar complaints: https://www.amazon.com/Joy-Abstraction-Exploration-Category-Theory/dp/1108477224/ref=mp_s_a_1_2?keywords=eugenia+cheng&qid=1695698839&sr=8-2#aw-udpv3-customer-reviews_feature_div https://www.amazon.com/Joy-Abstraction-Exploration-Category-...
  - shepherdjerred 3y ago
    > This is not just a mathematics book, but unfortunately also a polemic of progressive politics. For example, on p.42 there is a helpful definition of “cisgender”, in case you didn’t know. Then, on p.319, the notion of isomorphism — ubiquitous in mathematics, and really quite simple — is explained by considering two categories induced by a partial ordering, one consisting of concepts involving rich, white and male, and another involving rich, white and cisgender. It is then shown that rich white cisgender women have the same status in the second category as rich white males do in the first. Just below this, there is a helpful diagram showing that each of these categories is a 3-fold product of the category consisting of two objects and one arrow, namely (people with structural power —> people who are structurally oppressed). Really, there are many simpler ways to get these ideas across.
    - keiferski 3y ago
      Yeah, I try to avoid the culture war stuff, but this seems pretty unnecessary for a math book.
    - eru 3y ago
      She used the same example in some lecture on youtube, too.
4ad 3y ago
Yeah, I think the mathematical exposition was pretty good, but I couldn't bring myself to finish the book because of the culture war she kept trying to shove in.
- dagw 3y ago
  Are you referring to this book in particular or her writing in general?
  - 4ad 3y ago
    I am referring to her other book, The Joy of Abstraction.

smugma 3y ago

Is Math Real? Not when we are talking about how Math can be Complex.

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quickthrower2 3y ago
It can also be Rational. They hate to admit it, but it is all Imaginary.

moritzwarhier 3y ago

This introduction is a very pleasant read and adequately dense and easy at the same time.

tabtab 3y ago

Math is merely a mutual agreement. First an agreement on root principles, and then an agreement on notation for these root principles. The rest is built on top of those. Nobody is forced to accept the premises. (Unless you don't want an F in math class.)

dekhn 3y ago
One imagines two intelligent alien races would be able to communicate that pi is the ratio of the circumference to the diameter of a circle without coming to any real agreements over postulates or axioms. But I suppose that's natural philosophy, not math. See https://computingbiology.github.io/docs/hamming1998.pdf https://computingbiology.github.io/docs/hamming1998.pdf "Mathematics on a Distant Planet" for more thinking. Many physicists think that math is instrinsic and absolute (objective) in the universe, not "merely a mutual agreement"
- proamdev123 3y ago
  > One imagines two intelligent alien races would be able to communicate that pi is the ratio of the circumference to the diameter of a circle without coming to any real agreements over postulates or axioms. Except circles don’t exist in the real world. They are an abstraction that mankind created that are similar enough to many natural objects as to be a useful abstraction of them. In order for two alien races to communicate about pi, they’d have to first agree on the definition of a circle as a set of points equidistant from a central point. That’s one of the axioms they’d have to agree on before they could even get to pi.
  - jaza 3y ago
    Assuming that the two alien races are indeed "intelligent" (approximately relative to humans), and that they possess sensory perceptors approximately similar to those of humans, and that they inhabit the same observable universe as humans do - a universe that contains a great many physical (imperfect) circles - surely they should be expected to have conceived of the same abstraction of a (perfect) circle? And therefore to have gotten to pi? Then again, maybe a circle (and all other geometric abstractions) are too tied specifically to visual perception. Maybe, for example, intelligent aliens who only have auditory perception, would have no concept of a circle but would have gotten to the Doppler effect. Or intelligent aliens who only have tactile perception, would have no concept of a circle but would have gotten to the laws of thermodynamics.

A_D_E_P_T 3y ago

Mathematics is, in part, the study of universal traits: (1) Ratios between things in the world. (2) Logical relations between things in the world. (3) Absolute distinctions between things in the world, in a nominative sense, which is to say in the sense that numbers can be assigned to things or elements of things. Mathematics that relates to one of the three use-cases above is absolutely real, hence its unreasonable effectiveness in the natural sciences. (See Wigner: https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf ) When mathematics does not relate to one of the three -- for e.g., in Cantor's theories of "countable" and "uncountable" infinite sets -- it is totally unreal. A construct or game played with logic that has no prior or intrinsic relevance to the material world.

bawolff 3y ago
So when an application is found to a previously pure theory area (e.g. number theory and RSA) does math becone real? Is realness not an instrinsic property but just a judgement on how useful something is? Is being real a real property of something or just a construct?
- beezlebroxxxxxx 3y ago
  Math is in language. In a similar way that we can have fiction and non-fiction, with different uses of language, you can have 'pure' and 'applied' uses of math. Using 'real' as the goal kind of misses the point. It's not imaginative, if that's the antithesis of real in this whole conceptual discussion. But, like fiction to non-fiction, it's still a public phenomenon that "touches" the world in language as a shared, communal, element in a form of life. Math is remarkably useful as a representation of the world and for us solving problems, but you can also give directions to a location to someone with just hand signals and barking like a dog. In a way, math is not special. But, the degree of accuracy that we've developed in math, however, is what makes it remarkable.
  - bawolff 3y ago
    > Using 'real' as the goal kind of misses the point It misses the point of the question "is math real"? Tbf, i am actually sympathetic to the position that "realness" is not really relavent or a well defined term when it comes to what is basically a descriptive language of patterns and relations. But "is it real" is the question we started the thread on.

kdottt 3y ago

“The question “is math real?” is answered in the epilog of this book. Cheng tells us that math is real because it is an idea and ideas are real.” Alright great lol

dasil003 3y ago
What is “real”? Well, it’s just an idea… really
- dekhn 3y ago
  There is, almost certainly, an objective universe that exists outside of human minds, and it also seems likely that all of mathematics can be determined through objective mechanisms.
chongli 3y ago
This is an idea that goes back to Pythagoras and Plato. Think of a triangle. Now draw that triangle on paper. If you look closely enough, you'll see "imperfections" in the triangle you just drew. Now ask yourself: "how do I know this thing I just drew is imperfect? Where did the idea of a perfect triangle come from?" Plato would say the perfect triangle comes from the realm of "forms". This mystical place which is "more real" than "reality" because everything there is perfect and everything here is just a flawed approximation. Plato also said that this is the place where our souls go when we die, and we engage in "congress" with the forms and then return to earth, reincarnated. When we learn things, we aren't learning something new but actually recalling memories of the forms. This is why everyone knows what a perfect triangle is but no one has ever seen one in the physical world.
- dr_dshiv 3y ago
  Because the Platonic soul (psuche) is different from consciousness. Actually, Philolaus (first Pythagorean to write a book) said that conscious feelings come from the combination of the mathematical soul with the body. So, Pythagorean soul can be viewed as the set of logical or conceptual forms— which constitute a person, yet can also be passed on from person to person. That’s a very different kind of reincarnation…!

RagnarD 3y ago

You might as well ask "Are ideas real?" in the broader philosophic sense. Yes, they (with math as a subset) are real existents within consciousness. The question of mathematical ideas' connection to reality is epistemological. "Real reality" pertains to metaphysics - the externally perceivable stuff of reality that exists independent of our minds (as in, eternally prior to, and subsequent to, the existence of any humans or other beings with conceptual consciousness.) Math is an epistemological tool to abstract the causality of existence into a simplified structure amenable to consideration and manipulation by human consciousness. A quadratic equation can be used to approximate the coordinates of the trajectory of a thrown object in a gravity field. I think the really interesting question is why that particular equation reflects that trajectory. Arithmetic operators - multiplication and addition in that case - in a particular order, are approximating the causal operations of existence that are actually at work. I think of this as the philosophy of mathematics and much more should be done to investigate it.

coxmichael 3y ago
> Math is an epistemological tool to abstract the causality of existence into a simplified structure The mathematics to accurately predict or relay reality is still complex enough that it’s often beyond us. You’re right in that it’s a tool to understand, but if we’re using simplified math for simplified reality, is it really epistemological? As you suggest, math isn’t outside the boundary of philosophic investigation. It never was in the past, and I don’t think better approximations change that calculation.
- balder1991 3y ago
  You made me remember this article you might have read before: https://www.wired.com/story/our-machines-now-have-knowledge-well-never-understand/ https://www.wired.com/story/our-machines-now-have-knowledge-...
  - coxmichael 3y ago
    Thanks, I’ll give it a read

tracerbulletx 3y ago

Brains model patterns in signals, math could be described as the brain's language for expressing higher order manipulations on all those models with communicable symbols. So it's kind of both real and invented, because brains invent, and brains are also just real objects and any models they form came from somewhere.

User23 3y ago
The statement that brains model patterns rests on the supposition that patterns to be modeled exist.
- tracerbulletx 3y ago
  If there isn't then I guess we're all just talking to ourselves anyways so it doesn't really matter.
  - glennonymous 3y ago
    Am I the only person who hates it when people say “anyways” when they mean “anyway”?
    - bloak 3y ago
      Perhaps it's acceptable in moderation, like "ain't". It's needed for a rhyme in a well-known song by Billy Joel, for what it's worth. On the other hand, there is a different song by Billy Joel in which "anyway" is used for a rhyme. No, I'm not a Billy Joel expert or fanatic. I just happened to notice, all right?
    - pnut 3y ago
      That stray character doesn't change the meaning of the sentence, so yeah, stylistic complaints are subjective and beside the point. We're not in school.
- reuben364 3y ago
  The notion of existence is doing a lot of the leg work. A epistemological solipsist can say things "exist" when they have a model of their subjective experience in which things can exist. It gets real funky because your interpretation of qualia results in a model that includes the brain and thus a model of how you are interpreting things as a model in your model.

agumonkey 3y ago

is math memory ?

autokad 3y ago

> Oftentimes, math is taught as a set of rules and processes divined from an authority in which learners must memorize the rules and strictly follow the processes in order to satisfactorily find the one and only correct answer I was surprised later in life when the majority of people I talked to felt Math was this way (and a good thing). To them, math is a set of rules you learn to follow to the T and use them on other problems. For me, everything a Math teacher conveyed was more of a recommendation, a suggested tool that I could incorporate into my tool box. The methods they employed to solve problems were a matter of preference to me, rather than rigid rules. This way of thinking has always had some pros and cons. I never solved a problem the way a 'grader' was expecting. Some teachers loved the creativity and efficiency, other TA's just marked as zero. It made applying what I learned to other things, but on exams I would always be stressed with time because I would spend time on which way I was going to answer the problem. That being said, I am very good at Math. scored well in HS/college, 165/170 Math GRE score, in a field where Math is important (Data Scientist).

dingosity 3y ago

"Is Math Real?" Define real.

Cerium 3y ago
Not having an imaginary component.
- galangalalgol 3y ago
  I read the comments here just to verify I would find this joke.
  - sacnoradhq 3y ago
    ◻
    - quickthrower2 3y ago
      QED, for the oldies
- quickthrower2 3y ago
  And in addition a real number can encode an infinite amount of information, and as such real numbers are a construct that doesn't exist in the physical universe. Making them, unreal.

jacurtis 3y ago

This book is on my reading list. I listened to an interview of the author who said that their primary motivation of the book was less about teaching math, but teaching the value of looking at the world abstractly to determine patterns and constructs to explain our shared experience. Is math real or not? It doesn't matter she posits, it works and continues to work and we can learn from its existence that almost everything can be explained and is "predictable" given enough inputs.

itvision 3y ago

Really not the best title for this book.

conformist 3y ago

I've encountered this argument before, but also applied to other subject such as eg Latin: > 3) math has an indirect usefulness which is a way of thinking that is transferable to a myriad of disciplines and solutions to problems in everyday life. And it is this third reason that makes math relevant for most people. Are there actually any good studies that show this in a counterfactual setting? Like, do we actually know that spending time teaching maths (or, less plausibly, Latin) helps students aquire these abstract skills more than other subjects? Is this "transferability" of meta-skills a testable outcome?

JackFr 3y ago
I’m skeptical of (3) as well, and yet I encourage students to study math through calculus, and to study Latin. My feeling is that the utility of higher math, Latin, history and literature comes every minute of every day, as you experienced life as someone who has familiarity those things and your life will be richer and fuller. This is decidedly not testable. And yet I still believe it.
- fiforpg 3y ago
  That (3) is a very interesting issue. My suspicion is, it basically says "knowing logics is good for your well-being". The skill of logical inference — even at the level of very basic syllogisms — is both very much underappreciated and underdeveloped in the American college population, at least from my personal experience. As good citizens, we all collectively should grab a couple of Martin Gardner's or Lewis Carroll's books off the shelf and give them a good read. I predict it will do much good... and if I'm wrong, it certainly won't do any harm!
- m463 3y ago
  There are plenty of ways to use this stuff in everyday life. I took logic in college, and although the "logic as english statements" stuff was sort of confusing, the symbolic stuff like A&B = !A|!B stuff has helped with computers all my life. It was only much later in life that I ran back into logic as english statements in a way that made sense as practical. I read a book where they took apart the statement: If you loved me, you would take me to the movies. Most people in relationships will respond to this with: Well, I just took you to the movies last week! Why do you want to go again tonight, we had other plans! etc... But the book explained that with "If X, then Y" it was futile to address Y. You must address X: Wait, do you think I don't love you? Of COURSE I love you! ...just hard to unpack this in the middle of an emotional situation unless you've studied the logic :)

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ants_everywhere 3y ago

In my opinion, a more interesting question is "is anything other than math real?" Is there anything that physically exists that is non-mathematical?

the_af 3y ago
Well, what do you mean when you say something that "exists physically" is "mathematical"? Depending on your answer, the answer to your question could very well be "everything".
- ants_everywhere 3y ago
  For example, we could ask whether anything that we think of as "existing" would be missed by simulating the equations of physics with the empirical starting conditions of our universe. There are real limits in what humans know about the universe. But it's possible in theory that everything that is "real" follows from some differential equations.
  - the_af 3y ago
    In theory, maybe. I don't think our models of the universe are anything other than imperfect but useful approximations though. Could your question be rephrased as "take the real world, with its things we don't have an accurate model of (which is most of them). Would we notice if we replaced this universe by a tidier one, perfectly described by a math model?". It's an interesting, scifi question alright. I don't know if it has an answer.
cool_dude85 3y ago
What would it mean in this context to be "non-mathemtical"?
- ants_everywhere 3y ago
  I mean that it can't be completely described by mathematics

jjgreen 3y ago

... the typical working mathematician is a Platonist on weekdays and a formalist on Sundays Philip Davis, Reuben Hersh, "The Mathematical Experience"

User23 3y ago
Meanwhile the actual working mathematicians, better known as engineers, still use geometry on weekdays.
- eru 3y ago
  Engineers are their own special species. They aren't just 'working mathematicians'.

ftxbro 3y ago

some of it is integer

PrimeMcFly 3y ago

Yes. Next question.

fiforpg 3y ago

Curiously, this review (and perhaps the book being reviewed?) makes no mention of the model of mathematics as a language. This seems to be a common idea among working mathematicians and physicists[1,2], and it is rather clear that the question of reality of a language is ill-formed. A rich enough language should be suitable for expressing both real and abstract statements. It is also interesting to think about mathematics from the point of view of programming language paradigms: is it functional (I think mostly), what is its type system, etc. [1]: Galileo says that, roughly, the book of nature is written in the language of mathematics https://en.m.wikipedia.org/wiki/The_Assayer https://en.m.wikipedia.org/wiki/The_Assayer [2]: A classical essay by Wigner on the "unreasonable effectiveness of mathematics" also brings up the language analogy. https://web.archive.org/web/20210212111540/http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html https://web.archive.org/web/20210212111540/http://www.dartmo...

worldsayshi 3y ago
So language isn't mathematics? Like physics "is" mathematics? Are both language and math a third thing? Like maybe automata?
- winwang 3y ago
  considering that formal languages are part of math, there's little reason to believe natural language isn't. ft. LLMs.
- dexwiz 3y ago
  Mathematics is the language best suited to communicate Physics. > The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed. Translates to > F=ma
  - candiddevmike 3y ago
    Now translate that back to the original English without any context. I don't think your strawman is as succinct as you think it is.

sravic01 3y ago

Unfortunately, it takes extraordinary effort by the math teachers to make Mathematics interesting to students. Even if some of the teachers are willing to put the effort in teaching this wonderful subject in a way that makes sense to students, it's not possible to turn all mathematics teachers into teachers who understand how mathematics MUST be taught. I hope all these books, YouTube videos, and websites can help in making students curious about mathematics and explore further on their own. It is hopeless to even expect the school systems in US and a lot of other countries, to change the way their school systems work - teachers are not given enough time to spend on the topics in mathematics. It takes lot of extra effort by the students.

User23 3y ago
The actual reality is most students don’t have the cognitive horsepower for much past long division. I’ve seen adults struggle with basic algebra: “but what is X?” And so on. And that’s fine. Billions of people are going to live satisfying and fulfilling lives without ever knowing the chain rule. Edit: I thought it was pretty advertent.
- burnished 3y ago
  I think you inadvertently highlighted what I suspect is the real issue; a ruthless focus on the path to calculus to the detriment of other useful maths.
  - otabdeveloper4 3y ago
    Other maths are more difficult to grasp than calculus. Calculus at least has physical applications and can be visualized.
    - burnished 3y ago
      I think I disagree! Calculus is great and all (truly), but logic, boolean algebra, probability, statistics, even linear algebra all have really interesting and useful insights and are very approachable. I'm sure there are other maths I've never encountered that can also be enjoyed with minimal pre-requisite knowledge, though that might be more in the 'abstract and difficult' territory. The bits you need from calculus to approach those other subjects is also very minimal and approachable relative to the content of a calculus pre-req (where applicable).

GuB-42 3y ago

Math is rational, therefore math is real. Because ℚ ⊂ ℝ of course, but it is also not that far off from the conclusion of the book in that "math is real because it is an idea and ideas are real". In a non-mathematical sense, rational means "based on reason", and reason is the power of the mind.

tomrod 3y ago
I love your response. But can math be both rational AND irrational?
- elliotec 3y ago
  Yes. But does irrationality imply non-reality?
  - jacurtis 3y ago
    I work with several individuals who are consistently irrational and frustratingly real.
  - slowmovintarget 3y ago
    Math can also be imaginary, but this also seems to be a feature of the universe.
  - tomrod 3y ago
    Not for a numberphile.
- cgriswald 3y ago
  In some cases he's rational, irrational, but not real: https://en.wikipedia.org/wiki/Math_fab_Mathonwy https://en.wikipedia.org/wiki/Math_fab_Mathonwy
- jaza 3y ago
  Only if it's Schrödinger's math

beiller 3y ago

I think I heard that all numbers in the real world are irrational. So that means most of math is not real, except of course the irrationals like pi :)

adastra22 3y ago
That seems like a very weird statement to make. Many numbers in the real world are integral (two objects, one electron, etc.). Thanks to quantum physics, most measurements are integral too.
- beiller 3y ago
  Explained better above. There are more irrational numbers, almost guaranteeing any number come across in nature is irrational. Interesting thought since I think one thing that makes irrational numbers is there is no function for them. So it's kind of a cheeky way to say no math formulas can ever describe the real world since all the numbers are irrational.
  - dagw 3y ago
    There are more irrational numbers, almost guaranteeing any number come across in nature is irrational There is nothing that says that the distribution between rational and irrational numbers that show up in nature is the same as the distribution in our construction of the real numbers.
  - adastra22 3y ago
    You’re missing the point: physics has shown the world to be discretized. Almost every number you go out and measure IS ACTUALLY AN INTEGER. In that sense the real number system doesn’t exist. It’s super useful, yes, but it’s an abstraction away from reality.
    - 2snakes 3y ago
      I thought that "infinities of infinities" ala Cantor exist though. I agree with you that reality being quantized means everything in physical reality can be normalized to integers.

EpiMath 3y ago

This book is not really addressing the more common "is math real" question of it being empirical or invented. For an interesting take on that question, see the 1st section of the 2nd part of Daniel Shanks' Solved and Unsolved Problems in Number Theory. He makes some interesting points about the old Pythagorean views

dr_dshiv 3y ago
Pythagorean ideas—- well, it has been some of the most enriching philosophy I’ve ever encountered. It’s a rabbit hole, for sure.
- eru 3y ago
  It's interesting that mathematicians, when asked philosophically, might have all kinds of interesting and nuanced ideas about this topic. But when you let them get back to their mathematics, they behave as if they believed deep down in their heart that mathematics exists independently of the observer. (It's a working attitude that works well in practice. Just like a heliocentric world view works well enough for most celestial navigation you can do without computers.)
  - quickthrower2 3y ago
    I am waiting for the universe to do something non-mathematical. There is no reason to believe it can't :-)
  - mbork_pl 3y ago
    Sorry to break it to you, but most mathematicians don't give a s*t. Even worse, a large percentage of mathematicians (not sure if "most", but I'm afraid yes) do not usually have "interesting and nuanced ideas" (nor opinions) on anything.
hhs 3y ago
> This book is not really addressing the more common "is math real" question of it being empirical or invented. Please note, this is mentioned at the beginning of the review: "I settled in to read the book “Is Math Real?” expecting to become embroiled in the age-old controversary of whether math is invented or math is discovered. Instead, I found myself confronted with two viewpoints of mathematics: one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively. The premise of “Is Math Real?” is that people have different emotions about math. Some love the math and have little difficulty determining the correct answer to a problem while others loathe and dislike the math and have a difficult time ascertaining the correct response. Many times, a student is humbled or chastised for asking ‘a stupid question’. Author Cheng states that there are no stupid questions. In fact, the most profound concepts in mathematics are learned from asking the simplest of questions.”

olliej 3y ago

It is if you use pascal :D

fiforpg 3y ago
Do you mean Pascal the person? :P
- olliej 3y ago
  I was making a terrible joke about the old “real” type for floating point in pascal. Or possibly PASCAL :D

captaincrowbar 3y ago

Math is integer, unless declared real.

imchillyb 3y ago
Definitely found the Fortran guy.
- sacnoradhq 3y ago
  Double down on C.
sacnoradhq 3y ago
That's irrational and imaginary. Oh, and the cardinals will win the series.
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readyplayernull 3y ago

Well, think the other way around: if reality didn't keep the relationship between ratios as a coherent structure, it would be a soup of pure timeless randomness, producing lots of short lived Boltzmann's brains... so, we wouldn't notice.

didibus 3y ago
I remember reading of a theory that kind of said that in any large enough random set, over long periods of time, you can identify clusters which will have consistent and predictable behaviors, but only within the bounds of the cluster and for a set period of time, eventually drifting to something else. I thought there was mathematical backing of this observation. Which means it could all be just random, but we happen to exist at such a small scale within it all, and for such a short period, that we're able to find laws applicable to our pocket and our time period that appear to be true and can predict behavior. Thought it was interesting.
- ashbee 3y ago
  Definitely interesting. I dislike thinking about such things though. I've been stuck on this one for a while: Descartes posited, "I think, therefore I am" as a reassurance of our existence in a given moment. While this anchors our immediate awareness in some semblance of reality, it doesn't necessarily offer solace regarding the past or the future. Imagine, if you will, an expansive universe characterized by its sheer randomness and chaos, existing for eons. If, in such a universe, a consciousness can spontaneously emerge and dissolve—much like a fleeting set of physics—how can we be certain of its duration? Does it persist for a lifetime, or does it flicker for just a moment? Our memories might suggest a continuous existence, but what if they are merely ripples in this vast sea of randomness? It's an unsettling thought: could it be that we exist only in the fleeting now, borne from the precise alignment of countless variables just to perceive this very instant? And that perhaps, a moment ago or a moment hence, we simply weren't and won't be? While such thoughts might be disconcerting, their implications are, in a way, irrelevant. Even if I could unequivocally prove this transient nature of existence, our ephemeral consciousness would barely have time to grasp it. Our understanding would be obliterated, only to be replaced by another random state. And yet, if this fleeting moment is all we have, I'm honoured that you spent it reading this randomly generated comment. Not that you had much choice in the matter. I think it was Shakespeare who said: it was the best of times, it was the blurst of times.

drdec 3y ago

A better question is - is math an inherent part of the universe or is it a uniquely human construct?

otabdeveloper4 3y ago
It can be both.
mensetmanusman 3y ago
The universe has a mind (human) and its current opinion is that math is inherent and discovered.
rramadass 3y ago
Mathematics is an inherent part of the Universe. What we Discover/Invent are Models/Notation for identifying and using them. The difference today is that we have built and extended the Abstractions/Relationships to such an extent that people feel it is not "Real" which is of course not the case. The best argument for this are the various structures across the World (eg. Pyramids in Egypt/Central-Latin America, Temples/Structures in India, Aqueducts from ancient Rome, Great wall from ancient China etc.) spanning thousands of years which could not have been built without a knowledge of Mathematics as we define it today. Their approach and models/notation may have been different but the essence of the Mathematical Abstraction is the same.
- meroes 3y ago
  There’s so much more math though. More than will ever be seen or used in structures, thought about, etc even in principle. E.g. if string theory isn’t real the math still is, so where is it? Part of the universe is pretty unsatisfying.
  - rramadass 3y ago
    >There’s so much more math though. True, that is why I used the words "Discover/Invent". Also i used physical structures as an example since they are the most visible and unarguable evidence of "Real" Mathematics from the earliest times. You can invent abstractions to model concrete things(eg. all that is needed to model a skyscraper) or to model still further abstractions in a chain (eg. vector spaces for multidimensional/functional/etc. spaces). We only realize that it is "Real" when it is "Applied" in the concrete World (eg. number theory in cryptography).

drbojingle 3y ago

Is value real? Does it matter?

mensetmanusman 3y ago

Is it knowable if anything is real? What does knowable even mean?

russdill 3y ago
If math is real, nothing is real.
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fifticon 3y ago
in particular, what does 'mean' actually mean?

corethree 3y ago

Here's the irony. In science and therefore reality as we know it nothing can be proven to be true. Things can only be falsified in science. This occurs because if we make 10 million observations that verify a hypothesis we still haven't proven anything to be true because there always exists the possibility that a subsequent observation falsifies the entire hypothesis. While we can't prove anything in reality is real, we can prove things to be true in mathematics. Proof is the domain of math and logic not science. Therein lies the irony. We don't know if anything is real in science and therefore reality as we know it but we can verifiably know whether things are real in the universe of math. It really puts the question in perspective: what does it even mean to be real?

bandie91 3y ago
> what does it even mean to be real? exactly the question which has to be answered before answer "is X real?"
Warwolt 3y ago
But this just isn't the case, for your claim to be true we would have to be constrained to only do _inductive_ reasoning. When we do _deductive_ reasoning, and we feel as though the reasoning has been done correctly, then we know something _assuredly_. Mathematics and empirical science are very different in this respect, that much of mathematics is purely deductive.
- corethree 3y ago
  Inductive reasoning is by nature probabilistic so it can't prove anything. See: https://en.m.wikipedia.org/wiki/Inductive_reasoning https://en.m.wikipedia.org/wiki/Inductive_reasoning The first paragraph mentions this. As science is statistical in nature inductive reasoning by being probabilistic suffers from the same problem.
  - Warwolt 3y ago
    Your claim was that we can only know about reality through inductive reasoning, and this is the part I'm disputing, not that science itself is not inductive. Mathematical reasoning is deductive, and mathematics is part of reality.

keiferski 3y ago

A good starting point for questions like this is the SEP article: https://plato.stanford.edu/entries/philosophy-mathematics/ https://plato.stanford.edu/entries/philosophy-mathematics/

photon_lines 3y ago

If anyone doubts whether math is 'real' or not - maybe watching this clip of these 4 men might clarify it (pay attention to the language they use to discuss the architecture of the universe): https://youtu.be/AmUI2qf9uyo?si=6B6qA249jZqCVV5j https://youtu.be/AmUI2qf9uyo?si=6B6qA249jZqCVV5j

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redder23 3y ago

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slmjkdbtl 3y ago

How does the word "real" serve practical purpose, besides for the fun of logic / language game? It's a tool that we use and we found useful.

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WheelsAtLarge 3y ago

It's a tool. So it's as real as any other tool we have. It helps us bring about our thoughts in to the real world. We can't touch it, in the same way we can't touch language, but it's useful and that makes it real.

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SomethingNew2 3y ago

Based on the Amazon reviews, this doesn’t look like a very good book. “A Lot of Non-math Digressions” “Way too wordy and discursive. Boring format.” “It's hard to imagine that someone would write a book ostensibly about math that has so much personal, political, social, and just plain silly off-subject commentary. This book is so interspersed with that kind of commentary that it is difficult to focus on the math.” “The book has some helpful insights regarding teaching elementary students basic mathematical operations. However, it is filled with so many abrupt political digressions, often incoherent, and ironically, occasionally illogical, that the entire work is barely readable.”

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bowsamic 3y ago

I can already tell that a vast majority of the replies here are going to respond to the title and not the post

tempaway33751 3y ago
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crabmusket 3y ago
> The question “is math real?” is answered in the epilog of this book. Cheng tells us that math is real because it is an idea and ideas are real. Solved!
- bowsamic 3y ago
  This is not an alternative to people reading the article
  - crabmusket 3y ago
    Agreed. I just thought this little paragraph at the end of the article, which does directly address the title of the article, was amusing.
- Lockal 3y ago
  If ideas are real, do you have any idea about entities that are not real? I mean, what is his exact definition of "unreal" things? Something that can not be formally described?
  - crabmusket 3y ago
    I'll let Parmenides take that one > It needs must be that what can be spoken and thought is; for it is possible for it to be, and it is not possible for what is nothing to be.

nologic01 3y ago

> one view is that mathematics is a stiff and fixed set of rules and algorithms while the other view is that mathematics is flexible and our understanding of math comes from questioning of why mathematics functions so effectively I havent read the book but this dichotomy is not an intrinsic feature of how people pursued mathematics historically. The "stiffness" and excess focus on rigor and accuracy developed gradually over the 19th century because people were being loose canons - primarily around calculus.

leetcodesucks 3y ago
Its still not stiff enough. Mathematicians really ought to settle on a sound and coordinated syntax
- eru 3y ago
  Syntax is about the least of people's worries when they worry about 'mathematical rigor'. Math is made for people to read and write, and they have different 'domain specific languages' for different parts of math. Now what would be useful is a tool, perhaps something like a large language model, to automatically translate from the notation used in one area of math into another. That can't be a fully mechanical procedure (hence the need for something flexible like an LLM), because it's part and parcel of human mathematics to abuse notation here and there in the name of ergonomics. My personal pet peeve is defining a function like f(x) := x + 3, and then treating f(x) as the name of the function, instead of just f. But really, it's just a harmless abuse of notation when done by some humans to other humans.
  - rudy6912 3y ago
    I do not think mixing up 'f(x)' and 'f' is harmless, given mathematics is all about clarity. In any quality text, 'f(x)' denotes the value of 'f' at 'x', and 'f' denotes the function 'f' itself. Also, speaking of notation, I wonder why you used ':=' instead of '=' to define 'f'. There is no computation going on, right?
    - 3y ago

acec 3y ago

No if you use sqrt(-1)

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cdevries 3y ago

Everything is real by definition

DarkNova6 3y ago

No. Math is Complex.

auggierose 3y ago
No. Math is Surreal.
- anilakar 3y ago
  Elephant to the melting clockth power equals what?
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jongjong 3y ago

My view is that Math is real but the way it's represented is arbitrary. I'm sure that math could have looked very different than it does today and it would have worked better for a different kind of people. I find that some mathematical abstractions and ways of describing certain ideas are over-complicated and almost seem to be intended to confuse and obfuscate rather than to clearly explain. I feel similarly about certain fields of science; the ideas are really very simple but whenever you hear someone explaining them, the explanation seems to be full of gaps, rely on the audience to make certain assumptions, use custom terminology or terminology repurposed from different fields which can create confusion. I often felt that being good at math requires having a natural talent for making the right assumptions to fill in any gaps or ambiguity in explanations. I feel like the language of math does not do justice to the complex, intricate ideas it tries to convey. On the other hand, being good at programming requires the opposite; it's about being able to resist making assumptions.

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AvImd 3y ago
That's really interesting, can you give some examples?
- jongjong 3y ago
  I feel like almost all the descriptions of advanced (university level) math concepts are confusing when you first hear them. For example, the definition of a Taylor series according to Wikipedia: "In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point." This kind of description really throws me off. Especially "terms that are expressed in terms of the function's derivatives at a single point." first of all, superficially, the different meanings of the words 'term' and 'terms' throws me off and then even when I get over that, it's not clear what is meant 'in terms of' in what terms? What kind of relationship are we talking about here? I need to keep reading a lot more to fill that gap... In the meantime, I'm in a state of confusion and will need to re-read that sentence later to make sense of it once I have more info. I'm not sure how to solve that problem to explain it more clearly but it feels like definitions should gradually build up without gaps. Is it even solvable? I would prefer not having read that definition at all TBH as it only serves to confuse me, it has way too many possible interpretations. I prefer to jump straight to the formula.

ivanche 3y ago

I don't know if it's real but it's definitely complex.

ShamelessC 3y ago
Get out.
tannhaeuser 3y ago
OTOH it can also be rational and natural.
- thsksbd 3y ago
  And, amongst the reals, the irrational infinitely outnumber the wholesome
- ivanche 3y ago
  OTOOH sometimes it's irrational and transcedental so there's that.
otikik 3y ago
Came here looking for this comment, not disappointed.

kiriberty 3y ago

If math is not real, it should be complex then.