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The dirty secret of mathematics: We make it up as we go along (2018)
- morpheos137 4y agoMathematics may be made up in the sense of what we use for notation or symbology but the underlying relations are timeless and superuniversal. In any universe or species pi defined as the ratio of the path length traced by a set of coplanar points equidistant to a common point to the path traced by a set of points in a different plane intersecting exactly two points in the first path and the aforementioned common point will be the same as our pi.
- whycombinetor 4y agoIs this still true in hyperbolic geometry, where the circumference of a circle of radius r is greater than 2.pi.r?
- morpheos137 4y agoNever used the word circle in my definition of pi. If planes curve in another geometry then no. I think these would called surfaces rather than planes. You could define a plane as simply the set of points sharing a common normal vector not intersected more than once by lines oriented in directions that have a non-zero normal component (i.e. parallel planes are unique).
- red75prime 4y agoIn hyperbolic and elliptical geometry ratio of radius and circumference of a circle is not a constant. You can't define pi there as such. Anyway, pi comes up in various fields of math with no relation to the geometry of space we live in. Ironically, pi comes from geometry (land measurement), and the land they measured has elliptical geometry.
- ska 4y agoI think you are oversimplifying in a way that eludes the point OP was trying to make. Or rather two points. First that the process of actually creating mathematics is messy and largely made up as it goes along. I can create a new mathematical structure that turns out to not be very useful, etc. Secondly that the way math is taught typically hides this, and creates a very linear "greatest hits" approach which is misleading. You are correct that one of the things that has come out of centuries of studying mathematics are clear definitions of abstract objects that almost have to have been found; but the day-to-day isn't that. On the other hand, how something is taught and how it is practised often aren't that close to each other. Part of the reason the pedagogy looks the way it does is to distill centuries of thought and argument into a few credit hours.
- mr_toad 4y agoFor Pi to have any meaning requires a universe with at least two dimensions. Whether Pi would exist without a universe is a matter of debate.
- morpheos137 4y agoSo if I tree falls in a forest and nobody hears it it doesn't make a sound? Pi never has existed in a material sense. It exists as a logical potential consequent of abstract definitions regardless whether the universe does or not.
- mostlylurks 4y ago> For Pi to have any meaning requires a universe with at least two dimensions There is no such requirement. Mathematical constructs have meaning even outside of their most immediately obvious manifestation (such as the application of geometry to the understanding of physical space). Even a one-dimensional intelligent being, if such a thing is even possible, would eventually encounter pi, alongside the basics of euclidian geometry, as soon as they start doing math on datasets measuring more than one feature, as such datasets are embedded in n-dimensional spaces, where n is the number of independent features, even if physical space has less than n dimensions.
- smegsicle 4y agoif anyone gave a crap about your second sentence they'd point out that its impossible to parse without punctuation or rewording or something, but the fact that everyone already knows what you're saying despite an inscrutable explanation really pokes a hole in the idea that you are bringing up anything remotely novel
- jlokier 4y agoYour definition of pi gives incorrect values of pi... ("a set of points in a different plane" is not constrained to be a straight line.)
- robot_no_419 4y agoMath is presented in a way that's supposed to be organized, compact, and categorical. If we taught math the same way math was proven and discovered, it would be so slow and inefficient that we would still be covering linear algebra in post grad. As an analogy: The 1,000th person to climb Mt. Everest takes a well defined path that has already been mapped out as the most efficient path to the top. If every single person had to go through the treachery of finding the dead ends, cliffs, crevices, and death traps that the first few climbers endured, it would be a journey only a few could accomplish. Most people (computer scientists, engineers, chemists, physicists) using math only need to reach the top and see the view from the peak. The few climbers that are really dedicated to climbing (ie, the math researchers who reach the frontier of math) will naturally learn about the rest of the jagged, unmapped landscape as they climb harder and unconquered mountains.
- jacobolus 4y agoThe current pedagogy and curriculum is nowhere close to the “most efficient path” (nor is it the most intuitive, best organized, easiest to extend, ...). It’s just an arbitrary history-dependent path people happened to come up with, mostly centuries ago, and “refactoring” any of it is almost impossible. It’s more like our current public transit system: it gets some people where they’re going, somewhat on time, but it’s generally pretty crummy and full of historical inequity.
- throw7 4y ago"Columbus did not reach India but he discovered something quite interesting." What's going to bake your noodle is whether he discovered anything at all.
- ninkendo 4y agoThere are few useful definitions of “discover” that don’t apply to what Columbus did. Discover doesn’t have to mean you’re literally the first human to behold something…
- contravariant 4y agoDid he definitely know what he had found though? If you can discover something without knowing what it is then that opens up some interesting edge cases.
- ninkendo 4y agoIt’s only interesting if you’re interested in arbitrarily narrow, contrived definitions of words that everyone is already pretty clear on.
- tharkun__ 4y agoAccording to Marriam Webster: Definition of discover transitive verb to make known or visible : EXPOSE to obtain sight or knowledge of for the first time : FIND Even in other sources for the definition it is not necessary to either be the first or to 'understand' much of what you found/did. You simply have to have done 'it'. He discovered America as in he found it for the first time for the 'western world' of that time. Whether he thought it was India or connected to it does not seem to matter. He still found it and even found it first as to the knowledge of the Europeans alive at that time. It does not seem to matter that other Europeans (vikings) had also discovered it previously. Or that Asians had discovered it and stayed on the continent (native Americans) Can you elaborate on what edge cases you mean and why that would be relevant?
- tialaramex 4y ago
- dekhn 4y agoit's more like an undiscovered country. it was already there, we just didn't know about it. And sometimes, when we first find it, we think it's something else entirely.
- laichzeit0 4y agoIt's an open question as to whether mathematics is discovered or invented.
- dekhn 4y agoI'm definitely in the "math is discovered" after reading https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness... but clearly I have no proof.
- Quindecillion 4y agoThere's a really good book called Burn Math Class which highlights this fact very well, and actually helps with the learning process, making advanced math less intimidating.
- pdfernhout 4y agoThanks for this recommendation. I just ordered a copy.
- queuebert 4y agoThe Moore method attempts to address this. We used it in some courses at UT-Austin.
- paulpauper 4y agoThis is a taste of what mathematics looks like in practice. We make much of it up as we go. Sorry, but I strongly disagree. This is not true at all. Math, when done well, is based on a logical foundation that makes it understood by its intended audience. NO, they are not just making it up or blowing smoke. The apparent hand-waiving is still based on something which has a logical foundation or to save space.
- JadeNB 4y ago> Math, when done well, is based on a logical foundation that makes it understood by its intended audience. NO, they are not just making it up or blowing smoke. I think you may be misunderstanding what the author is trying to say, or, rather, what part of a mathematician's activity they're trying to discuss. Mathematicians are not making it up, in the sense that we're not engaged in some elaborate game with shifting but meaningless rules. The final product of mathematics is fully as polished and rigorous as its weighty presentation suggests. However, the act of doing mathematics is, or can be, messy, non-rigorous, and nonsensical on the way to discovering that perfect polish and rigor. Much like the old adage that, if you never miss a flight, you're spending too much time at the airport—if every bit of a mathematician's work is up to the full stands of rigor required of a final publication, then that mathematician is not doing the best work that they could do.
- nitwit005 4y agoThere are history of math courses. I took one and enjoyed it, but it's a vast topic, reaching into ancient history. It's unrealistic to cover much of it. And on top of that vast pile of material, there is the history of how teaching math has changed over time. My professor had some early US textbooks.
- cardamomo 4y agoI'm an elementary school math teacher and would love to learn more about this! Is there anything you can share about this course (syllabus, etc.)? Email address is in my profile
- nitwit005 4y agoIt's been too long. It would have been this course: https://catalog.sonoma.edu/preview_course_nopop.php?catoid=6&coid=32648 https://catalog.sonoma.edu/preview_course_nopop.php?catoid=6... There are some other courses you can find by searching "History of mathematics syllabus", such as this Berkeley one that has quite a few usable links and references: https://math.berkeley.edu/~wodzicki/160/ https://math.berkeley.edu/~wodzicki/160/
- plaguepilled 4y agoThis might be rude but... No shit? Maths is a deductive process, as in, you PRODUCE an answer given the problem. It necessitates 'coming up with' an answer. I think the author is making a fundamental confusion between maths as a class taken in school or uni, and maths as an intellectual discipline. The class is dogmatic and over structured, but that's due to administration pressures, not because mathematics teachers aren't aware of the issue. The discipline, while still adhering to structure, allows for reframing the problem and always has.
- JadeNB 4y ago> I think the author is making a fundamental confusion between maths as a class taken in school or uni, and maths as an intellectual discipline. The class is dogmatic and over structured, but that's due to administration pressures, not because mathematics teachers aren't aware of the issue. The discipline, while still adhering to structure, allows for reframing the problem and always has. I think the author is not making that confusion, but rather trying to help remedy that confusion on the part of a student in a school or university class who hasn't had anyone to spell out this fact (that every mathematician knows—but not every non-mathematician!).
- JadeNB 4y agoI appreciate these kinds of exposés, and students should know these things, but I'm not sure why one would ever assume otherwise. (That is not to say, I know that people do, but I don't know why.) Novels are presented to us in tidy, precise, packaged form—well, there are some novels that experiment with the form, but certainly it's the norm—but no-one feels the need to explain that a novelist is making it up as they go along. What's different between math and English—well, a lot, obviously, but specifically, what's the difference that causes non-practitioners to assume that the former is eternal and fixed, whereas the latter is a product of human creativity?
- motohagiography 4y agoIt's not a secret. It's what makes it so impenetrably annoying that the smartest people move on to problems they can have more of an impact on, casting suspicion on anyone who survives in the current establishment. There was a post the other day from stopa.io about implemeting Russel and Whitehead's "Principia" in LISP, and then describing Godel's theorems relative to it, that to me was the future of maths. I'd posit that math is in the midst of the equivalent of the AI winter of the 80's and 90's, where current establishment thinking will be subsumed by practitioners just making shit work as the result of a generation of kids with access to things like Coq, theorem provers, and a bunch of hackers using functional languages, just like "AI" happened as the result of some adequate tooling available to people without gatekeepers. I think we're on the precipice of a rennaisance.
- ravi-delia 4y agoThere's been a ton of great work in math lately! I mean in number theory alone we have tools which would have made the greats of the past weep with joy, and we've only just begun to scratch the surface of what they can do. Implementing a formalism that frankly does not have a significant impact on any area of math which isn't directly based on working with it is not a sign that math as a whole is trending towards programming. Theorem solvers are, obviously, getting more awesome by the day though. Not quite there yet, but damn are they close
- motohagiography 4y agoIndeed, it's not that math is trending toward programming, but programmers are trending toward math, which means more minds on the problems acquiring the tools. It's exciting to read about too. Quanta has some of the best writing I've ever read anywhere. I am a layman who is enthusiastic about it and sees the most important problem to solve as being how to scale to new minds faster. The most influential people in my own life were not the best at what they did, but the ones who let on that the bar to doing those things and being a part of the conversation was closer and more achievable than the headlines about virtuosos made it appear. We're only ever 3-5 years from learning anything, imo, and I think a lot of the opportunity to have an impact will be in writing the next GEB, Chaos, Emperor's New Mind, or other popular treatment, where instead of focusing on and solving one problem, we can inspire and apply a million new minds to several of them.
- kzz102 4y agoThere's nothing wrong with presenting existing knowledge in the most organized, clear, and accessible way. Historical context is useful, but I don't think we need to spend a lot of time explaining the initial confusions. When you try to discover new mathematics, it's always messy. But if you want to learn about discovering new mathematics, you should just try to do a research project, instead of trying to learn already known stuff pretending you are discovering it for the first time.
- perilunar 4y agoSee also: A Mathematician’s Lament, by Paul Lockhart https://www.maa.org/external_archive/devlin/LockhartsLament.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... https://en.wikipedia.org/wiki/A_Mathematician%27s_Lament https://en.wikipedia.org/wiki/A_Mathematician%27s_Lament
- joshgree88888 4y agoHe wrote a second book called "Measurement" which I loved!
- revskill 4y agoI found advanced maths is like learning engineering. They're less fun than junior maths, where it triggers thinking more effieciently. So to me, the secret here, is advanced maths is represented boringly on purpose mathematicians (like a note), instead of education purpose.
- amelius 4y agoAdvanced maths adds layers of boring bookkeeping you have to wade through before you get to anything interesting.
- Sir_Liigmaz 4y agoThe spells are increasingly complex, but quite powerful.
- red75prime 4y agoThe dirty secret of every activity: we make mistakes and fix them. Mathematics is no exception.
- n4r9 4y agoReminds me of a joke I once heard. A group of university dons are discussing next year's budget and which subjects can cut down on equipment. The physicist says "I'm sorry but we need interferometers, telescopes, and lasers to teach anything". The chemist says "Hah! All we need is some bunsen burners, test tubes and chemicalsto run experiments" The mathematician says "Pfft! All we need is pen and paper to explore ideas, and a trash can for when the idea turns out to be wrong." The philosopher says "Well, all we need is a pen and paper".
- sidkshatriya 4y agoThe content of the article is good. I do feel the title is click-baity in my opinion. This was not necessary. Progress in any field of endeavour is never a smooth process, I think most people who know a bit of science/math know that. This is not "making it up" as we go along, rather I would say it is the usual process of experimentation. This is the normal process of discovery and the article explains how math progresses in often non-monotonic, discontinuous steps. So, once again, good article with good content but I wish there was some other title to it.
- CogitoCogito 4y agoThis post isn't really that deep. The writer's discovery of the real nature of math is in line with essentially all disciplines. Things are usually presented as more rigid to make learning easier, but in all fields of research the lines are much more blurred. But yeah sure it's good that more people understand this. When I used to teach math to university students, this true nature of math is something I always tried to get across. Students would tend to fight this by nature (possibly because they feel more comfortable in a world where mathematical "laws" truly are laws), but I usually pushed back since I thought the insight was important. I'm not entirely sure it was though.