Was Mathematics Invented or Discovered?
Date: 01/01/2001 at 15:37:07 From: Angela Subject: Was mathematics invented or discovered? Dr. Math, I'm currently taking a course in high school that studies (along with philosophy and logic) mathematics. We have been assigned to write a 1,000-word essay answering the question "was mathematics invented or discovered?" Can you please help me with this difficult question? From my point of view, it can be argued either way. I realize that you addressed this question back in 1995, but I think it is a question that deserves to be elaborated on. Or, if not, could you direct me to some Web sites that will give me different viewpoints as to the history of mathematics? I personally think that the concept of mathematics was discovered. You can't "invent" 1 + 1 = 2; it just does. I'm confused as how to present my viewpoint. What other possible arguments are there? It will be very helpful if you could give me ideas from both sides. Thank you very much for your help. It will be greatly appreciated. Angela
Date: 01/02/2001 at 13:59:32 From: Doctor Ian Subject: Re: Was mathematics invented or discovered? Hi Angela, Let's begin with your assertion that "You can't 'invent' 1 + 1 = 2; it just does." On the contrary, 1 + 1 = 0 in mod(2) arithmetic 1 + 1 = 10 in base(2) arithmetic 1 + 1 = 1 in Boolean arithmetic 1 + 1 = [an arbitrary value] in group theory, depending on your choice of group. In fact, '1 + 1 = 2' is a consequence of selecting one particular set of rules to define one particular formal system. But there are lots of other systems that can be defined, and confusing 'Peano arithmetic' with 'mathematics' is like confusing 'Chess' with 'games'. You should probably take a look at the following discussion from the Dr. Math archives: Understanding Mathematics http://mathforum.org/dr.math/problems/nwasokwa.7.25.99.html But you should also give some thought to whether 'invented' vs. 'discovered' is a distinction without a difference. Does the answer to any other question depend on the answer to this one? Would it affect how mathematics is created, or used? Would there be any discernible difference between a world in which mathematics is invented and one in which it is discovered? To get a little practice thinking about this kind of thing, consider the question: do human beings have free will? Is there any outcome or situation that can be explained by free will that can't also be explained by predestination, and vice versa? There isn't. In which case, what is the point of asking the question? (If you don't understand what I'm getting at, find a copy of the book _Pragmatism_ by William James and read it. In fact, you should read it anyway. It will save you a lot of useless head-scratching over the course of your life.) Personally, I think that the question 'Is math invented or discovered?' is meaningless. But thinking about _why_ it's meaningless can lead you to some very interesting ideas about the nature of meaning! (For example, if you're lucky it might lead you to read Doug Hofstadter's wonderful essay 'A Conversation with Einstein's Brain', in his book _The Mind's I_.) So I hope you'll continue to noodle around with this issue this even after your essay has been completed. I hope this helps. Feel free to write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 01/03/2001 at 17:26:17 From: Angela Subject: Re: Was mathematics invented or discovered? Hi Dr. Ian: Thank you so much for your quick response. Although I agree with your response that "the question 'Is math invented or discovered?' is meaningless" I have no choice! I have to write the essay anyway. And it will be difficult to come up with 1,000 words on this topic. I read the discussion that you suggested, but I am still confused. So, I am writing back to ask you some more questions. Here goes! 1) What exactly is Peano mathematics? 2) What distinction, if any, is there between 'invented' and 'discovered' with regard to mathematics? Although I agree that it makes no difference, for the sake of the essay I must come up with some logical arguments. I think that the concept of mathematics, as a whole, was discovered. I don't think that it could be invented. However, I can't put into words why I think this; it's very difficult to explain. What do you think? 3) Is it possible that 'invented' and 'discovered' are essentially the same thing? 4) Goedel was mentioned in the discussion. What philosophy did most mathematicians have when faced with this question? 5) Where can I find more information or background on mathematics and philosophy? Thanks again for your help, Angela
Date: 01/03/2001 at 19:10:43 From: Doctor Ian Subject: Re: Was mathematics invented or discovered? Hi Angela. You're very welcome. >Although I agree with your response that "the question 'Is math >invented or discovered?' is meaningless" I have no choice! I have to >write the essay anyway. Yeah, I understand that. I was just trying to give you some options for making the best of a bad situation. As a friend of mine likes to say, 'If life hands you a lemon, build a high-pressure explosive gas powered lemon gun and blast it to oblivion'. :^D In other words, when given an assignment like this, I've often had success analyzing the assignment itself, rather than just submitting to the tyranny of educational orthodoxy. But I realize that this approach isn't for everyone. >1) What exactly is Peano mathematics? Peano mathematics is a game defined by the rules described here: Peano's Axioms - Carl Lee, Dept. of Mathematics, Univ. of Kentucky http://www.ms.uky.edu/~lee/ma502/notes2/node7.html It's what you normally think of as 'arithmetic', but it's just one possible _kind_ of arithmetic. >2) What distinction, if any, is there between 'invented', and > 'discovered' with regard to mathematics? I honestly don't know if there _is_ a distinction... or, more to the point, I don't know that you could ever prove that one exists. You can't really even begin looking for the distinction until you answer the prior question of what constitutes mathematics. Is mathematics a process? Is it a collection of facts, or patterns, or theorems? Would mathematics continue to exist if all conscious beings were eliminated from the universe? Do the theorems that we haven't yet worked out already exist, in some implicit form? Or do they come into being only when they become known? This is why I think you should read 'A conversation with Einstein's brain'. It's pretty short, wonderfully written, and it will give you a _lot_ to think about with respect to this issue. (That way, regardless of how the essay comes out, the experience won't have been a complete waste of your time.) There is a great scene in Kurt Vonnegut's book, _Cat's Cradle_, in which a hard-core scientist is arguing with his secretary about truth. He challenges her to say something that is absolutely true, and she responds by saying "God is love." He looks at her and asks, "What is 'God'? What is 'love'?" This is the essence of philosophical exposition, as practiced in the West. You choose two words whose definitions aren't agreed upon by anyone, and then proceed to claim that there is some necessary relationship between their referents: Math is [invented | discovered]. What is 'math'? What is 'invented'? What is 'discovered'? >Although I agree that it makes no difference, for the sake of the >essay I must come up with some logical arguments. I think that the >concept of mathematics, as a whole, was discovered. I don't think >that it could be invented. What is 'invented'? What is 'discovered'? :^D >However, I can't put into words why I think this; it's very difficult >to explain. What do you think? I think you should read Richard Mitchell's book _The Gift of Fire_, see: Richard Mitchell's The Gift of Fire http://www.sourcetext.com/grammarian/gift-of-fire/ I think you should read it _today_. (It's online, so you don't even need to go find a copy.) >3) Is it possible that 'invented' and 'discovered' are essentially > the same thing? That's an excellent question. If there is an essential difference between 'discovery' and 'invention', would it perhaps be that something discovered has an independent existence, while something invented does not? Something uninvented doesn't exist, but something undiscovered does? (What is 'exist'? What is 'is'?) Are books invented or discovered? I could write a computer program that, given enough time, could generate every possible sequence of characters, so in a sense, every possible book that could be written is just waiting to be found, right? What is the _essential_ difference between a book that I haven't yet found because my program hasn't been running long enough, and a reservoir of oil that I haven't yet found because my drill hasn't been running long enough? Could it be that discovery implies physical existence? That would seem to be a convenient way out, but math _books_ have physical existence, and the patterns of neural activity that are formed in your head as you think about mathematics have physical existence. So you're not going to get off _that_ easily. :^D If you want to finesse the question completely, you might argue that mathematics is the process of inventing formal systems, in which theorems are discovered. I say that this 'finesses' the question, because it utterly fails to address the prior question of whether those formal systems are somehow guaranteed to emerge, given the structure of brains, in which case the systems themselves are actually discovered, rather than invented - but your teacher might not notice that. There is one other small item that you should definitely read. Find a copy of the book _Surely You're Joking, Mr. Feynman_, and read the chapter called... I think it's called 'A Map of the Cat'. Check the index of the book for 'Whitehead' or 'essential object', and you should find the part I'm thinking of. You'll see why I'm recommending it as soon as you've found it. (When you're done with the essay, go ahead and read the whole book.) >4) Goedel was mentioned in the discussion. What philosophy did most > mathematicians have when faced with this question? Most mathematicians have never considered this question, except possibly when drunk, or at freshman bull sessions in college. They're too busy inventing mathematics to worry about the possibility that they might actually be discovering it instead... or too busy discovering it to worry that they might really be inventing it. I forget which. >5) Where can I find more information or background on mathematics and > philosophy? If I were you, I'd probably go to the Google search engine at: http://www.google.com/ and search on the keywords 'math philosophy'. Reuben Hersh has written a book called _What is Mathematics, Really?_ that you might find helpful, at least as a source that you can quote. (Quotations count towards the 1000 word minimum, right?) He takes the 'Platonist' view, which is the idea that "mathematical entities exist outside space and time, outside thought and matter, in an abstract realm independent of any consciousness, individual or social." But if you want to take that idea seriously, you have to be willing to accept a pretty odd definition of 'exists'. (By his definition, unicorns 'exist' too. In fact, by his definition, everything exists, even if it doesn't.) My personal opinion is that Hersh doesn't sufficiently appreciate the difference between 'existence' and 'implication', but that's something you have to decide for yourself. And, as I suggested, you should really read James's book _Pragmatism_. Good luck with your essay. If you're finding what I'm saying at all helpful, please, feel free to continue our discussion. I'm rather enjoying it. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 01/04/2001 at 02:45:18 From: Angela Subject: Re: Was mathematics invented or discovered? Hello Dr. Ian: Again, thank you so much for your quick response. I was delighted to receive another message so soon. By the way, I agree with what your friend said about the lemons. I am somewhat taking the advice, while still operating within the boundaries of the requirements for this particular essay. I've found that my teacher rather enjoys analysis of any kind, related to the topic or otherwise. But unlike you, I feel I must partially submit to this "educational orthodoxy"! Your response generated a lot of ideas that I can use in my essay. I especially enjoyed your suggestion to "finesse the question." In fact, I think that I will incorporate that into my closing paragraph. Upon doing more research, I discovered that I must stand on the so-called "absolutist" side of the argument. I think that mathematics is universal and certain. Mathematics is discovered by the intuition of the mathematician, and then established in a proof. Do you consider yourself a mathematician? What do you think about this viewpoint? I also think that mathematics is woven into practically every aspect of the world, right down to nature. I'm sure you are familiar with Fibonacci numbers... however, I'm confused as to the exact name of the sequences, related to the Fibonacci numbers, that occur in nature. Any ideas? The opposing view to the absolutist is the "fallibilist" who argue that mathematics is invented and is revisable and changing. They say that mathematical truths are invented, or are by-products of inventions. Although I feel this is partly true, the concept of mathematics as it relates to everyday application, was not invented. What do you have to say about this? You commented: >Is mathematics a process? Is it a collection of facts, or patterns, >or theorems? Would mathematics continue to exist if all conscious >beings were eliminated from the universe? Do the theorems that we >haven't yet worked out already exist, in some implicit form? Or do >they come into being only when they become known? I think mathematics is all of these things, and that it would continue to exist even without conscious beings. The theorems exist as well, they just need to be discovered... they are realized when they become known, but in actuality they are waiting to be found. Is this wrong? What have you heard in your experience with mathematics in general? I have not yet read 'A conversation with Einstein's brain'. But I will before I turn in my essay. Furthermore, what do the words 'invention' and 'discovery' mean to you? Were there any mathematicians who where also philosophers? I am taking into careful consideration everything you say. All this information has provided me with a deluge of topics to discuss in my essay. Thank you for all the suggestions. I will try to locate the books you mentioned. Thanks, Angela
Date: 01/04/2001 at 15:33:53 From: Doctor Ian Subject: Re: Was mathematics invented or discovered? Hi Angela, >I agree with what your friend said about the lemons. I am somewhat >taking the advice, while still operating within the boundaries of the >requirements for this particular essay. I've found that my teacher >rather enjoys analysis of any kind, related to the topic or >otherwise. But unlike you, I feel I must partially submit to this >"educational orthodoxy"! I understand completely, and I sympathize. It's certainly much easier to take my view of things once the whole matter of trying to get into a top university is no longer looming in the future. :^D >Upon doing more research, I discovered that I must stand on the so- >called "absolutist" side of the argument. I think that mathematics is >universal and certain. You might want to look into something called the Axiom of Choice, (an introduction and links collection by Eric Schechter of Vanderbilt University) at: http://www.math.vanderbilt.edu/~schectex/ccc/choice.html Although the reference I'm citing here indicates that the axiom is universally accepted, I can tell you from personal experience that there are many mathematicians who feel that the axiom of choice has no place in mathematics, and refuse to accept any theorems that can only be proved by using the axiom. So what this means is that right here on earth, we have at least two sets of mathematicians who disagree on how mathematics may properly be carried out. Wouldn't it be a failure of imagination to assume that all mathematicians elsewhere in the universe would necessarily fall into one of these two camps? Animals, for example, may take a different view of things. Read, for example _What the Tortoise Said to Achilles_ by Lewis Carroll at: http://www.ditext.com/carroll/tortoise.html >Mathematics is discovered by the intuition of the mathematician, and >then established in a proof. Sometimes. One of the things Goedel showed is that in any internally consistent formal system that can be created, there will always be statements that are true, but that are not provable within the system. So, does that mean that these unprovable truths are somehow outside of mathematics? Or does it mean that your view of what constitutes mathematics may be too limited? But more to the point, why is it 'discovery' when the intuition of a mathematician comes up with a new way of doing things, but 'invention' when the intuition of an engineer does the same thing? Why wouldn't it be just as correct to say "Mathematics is invented by the intuition of the mathematician, and then established in a proof." If you want to conclude that theorems are discovered rather than invented, then to be logically consistent you would also have to conclude that endgame analyses in chess are discovered rather than invented. But that seems like a silly conclusion, doesn't it? By the way, from the way you've phrased your statement, it sounds like you're using the word 'mathematics' to mean the _theorems_ that are produced by mathematicians, rather than the process of deriving those theorems. Is that correct? If so, you need to explain how '1 + 1 = 2' and '1 + 1 = 1' can both exist within 'mathematics'. The answer, of course, is that although the two statements use the same symbols, the symbols have different meanings in the two statements. But that would imply that the meanings attached to mathematical symbols are arbitrary. And if that's true, how can math be 'discovered'? >Do you consider yourself a mathematician? I consider myself a scientist. >What do you think about this viewpoint? As I said earlier, I think mathematical absolutism does violence to the concept of existence by failing to distinguish between existence and implication. We have the word 'implied' to describe a class of concepts that might someday exist but do not yet exist. It seems silly not to use it. >I also think that mathematics is woven into practically every aspect >of the world, right down to nature. I'm sure you are familiar with >Fibonacci numbers... however, I'm confused as to the exact name of >the sequences, related to the Fibonacci numbers, that occur in >nature. Any ideas? Math is a framework for creating models. Sometimes the models have close correspondences with the objects and events that we observe in the world, and sometimes they don't. But it's crucial to realize also that the models we create determine which objects and events we observe! You've surely heard the joke about the drunk who looked for his car keys under a lamppost a block away from where he dropped them, 'because the light is better there'. In addition to its primary nature as a game, math provides a lamppost under which scientists search for theories to describe the world. But the only theories that they're going to find will be the ones under the lamppost. To claim that mathematics is woven into every aspect of nature is a little like looking at a car seat and concluding that every part of a car is made from leather. >The opposing view to the absolutist is the "fallibilist" who argue >that mathematics is invented and is revisable and changing. They say >that mathematical truths are invented, or are by-products of >inventions. Although I feel this is partly true, the concept of >mathematics as it relates to everyday application, was not invented. >What do you have to say about this? I think that's _one_ opposing view. Personally, I'm still trying to figure out what the essential difference between 'invented' and 'discovered' is. I haven't been able to identify one. >You commented: > >>Is mathematics a process? Is it a collection of facts, or patterns, >>or theorems? Would mathematics continue to exist if all conscious >>beings were eliminated from the universe? Do the theorems that we >>haven't yet worked out already exist, in some implicit form? Or do >>they come into being only when they become known? > >I think mathematics is all of these things If you think that mathematics is a process, and if processes are, by definition, arbitrary, then how can mathematics be discovered? >...and that it would continue to exist even without conscious beings. >The theorems exist as well, they just need to be discovered... they >are realized when they become known, but in actuality they are >waiting to be found. Is this wrong? Whether it's 'wrong' depends on how you define 'exists'. If a theorem that hasn't yet been articulated can 'exist' in the absence of any being that can possibly understand the theorem, then I no longer know how to complete the following sentence in a sensible way: _______ does not exist. Which is to say, 'exists' ceases to have any meaning at all. To say 'x exists' conveys no more meaning than to say that 'x zorbles' or 'x blinmarges'. But perhaps you can find something to put in the blank? (Be careful, because as soon as you stick something in there, I'm going to say that it 'exists' in some realm independent of space and time and physical causality.) I'm not sure whether he ever really said it, but President Nixon was often quoted as saying that the United States might have to 'destroy Cambodia in order to save it'. I think that the absolutist position destroys the concept of existence in order to 'save' it - that is, in order to claim that it applies to mathematical concepts. >Furthermore, what do the words 'invention' and 'discovery' mean to >you? As far as I can tell, the only practical difference is that you can patent inventions, but not discoveries. But that's a legal distinction, not a semantic one. I'm not sure that there is a 'bright line' (to borrow a legal term) that can be drawn between the two. If I wanted to try to create a hard distinction, I might try this one: Discovery applies to situations in which you don't get to choose the rules, while invention does not. So you can discover a law of nature, but you must invent a new technology in the context of other technologies that have already been invented. Another way to say this would be to say discovery is context-free, while invention is not. If you accept this definition, then the answer to the 'invention or discovery' question depends on whether cognition is context-free - that is, whether there is only one set of rules governing all possible modes of thought. Which hardly seems like a simpler question! (It's like the old joke: How do you put four elephants in a Volkswagen? Two in the front seat, and two in the back!) And even if the answer to that question turns out to be 'yes', then you also have to consider whether cognition is inevitable, or accidental. So I guess I would say that if there is only one set of rules governing cognition, and if the rules of the physical universe make cognition inevitable, then math is discovered. Otherwise, it is invented. But I might be able to talk myself out of that position. :^D >Where there any mathematicians who were also philosophers? Richard Feynman, for sure. And Archimedes. >I am taking into careful consideration everything you say. All this >information has provided me with a deluge of topics to discuss in my >essay. I'll bet '1,000 words' never seemed so small before! As I said, I'm enjoying this. If you'd like to talk about any of the books, feel free to write back. Good luck with your essay. I'd love to read it when you're done. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
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