# Mathematics: Invented or Discovered? **URL:** \ **Category:** Great Debates\ **Created:** [April 24, 2001, 2:26am UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084 "2001-04-24T02:26:44Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Psi\_Cop](https://avatars.discourse-cdn.com/v4/letter/p/c57346/32.png) [@Psi\_Cop](https://boards.straightdope.com/u/Psi_Cop)\ **Post date:** [April 24, 2001, 2:26am UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/1 "2001-04-24T02:26:44Z")
Several days ago, I got into a debate with some friends about whether we invented mathematics or discovered it. Most of them seem to have gravitated towards the “discovered” side, while I am left alone with only a few allies to defend the “invented” side. I’ve used this board for referencing other subjects we’ve argued about, but I cannot turn up any related threads by searching for keywords like “math,” “discovered,” and “invented.” So I present this question: Was mathematics invented or discovered by humans? Briefly, here are the main points of each side so far. I cite examples such as logarithms; the way it was once explained by a teacher was along the lines of “…and here, mathematicians were stuck on how to proceed. So they made up a new method.” I also use right triangle trig ratios. What’s special about a 90 degree angle in nature? Some of the other side’s arguments are based on the natural world having many patterns, such as the multiplication rate of bacteria. All of classical physics is also a sticking point – if we invented mathematics, did we also invent physics? Or just invent the way to interpret physics? So, what are the thoughts of the board members? -Psi Cop ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [April 24, 2001, 3:21am UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/2 "2001-04-24T03:21:20Z")
[My apologies if this appears twice, I’m getting error messages from the server when I try to submit this…] That’s a tricky question to answer until you distinguish between math as “mathematical properties of the real world” and math as “formal systems of mathematics”. As an example of the former, consider the phenomenon of “one”, i.e. unity. We certainly didn’t invent unity. But on the other hand we did invent the formal system known as arithmetic. Specifically, (if I may over-simplify for a moment here) the mathematician Peano set down the axioms and rules of arithmetic. Those axioms and rules together make up a formal system which _appears to describe_ certain natural phenomena like “one”, “zero”, and “addition”, but the formal system is not the same thing as the natural phenomena. In other words, you and your friends are each partly correct: we invented a language to describe things that we discovered. In the same way we invented the colour wheel as one way of describing colours, and Newtonian mechanics as one way of describing the motion of the planets. ---
**Author:** ![ITR\_champion](https://avatars.discourse-cdn.com/v4/letter/i/c67d28/32.png) [@ITR\_champion](https://boards.straightdope.com/u/ITR_champion)\ **Post date:** [April 24, 2001, 3:46am UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/3 "2001-04-24T03:46:22Z")
I’m going to cast my vote for discovered. Although the development of mathematics could have gone slightly differently, the general principles are set and can’t be altered. Take the specific example of the 90 degree angle. Right angles have a lot of important occurences in “the real world”, so I doubt that any mathematical system could exist without giving them special signifigance. In force fields such as gravity or magnetic fields, you do no work moving a particle in the field if and only if you move the particle at right angles to the field. ---
**Author:** ![Alessan](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/alessan/32/465_2.png) [@Alessan](https://boards.straightdope.com/u/Alessan)\ **Post date:** [April 24, 2001, 4:21am UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/4 "2001-04-24T04:21:55Z")
> [@](#): > > \*Originally posted by ITR champion \* > \*\* > Take the specific example of the 90 degree angle. Right angles have a lot of important occurences in “the real world”, so I doubt that any mathematical system could exist without giving them special signifigance. In force fields such as gravity or magnetic fields, you do no work moving a particle in the field if and only if you move the particle at right angles to the field. \*\* Plus, if you put a wall up at any angle other than 90 degrees from the ground, it’ll fall over. ---
**Author:** ![Nen](https://avatars.discourse-cdn.com/v4/letter/n/c5a1d2/32.png) [@Nen](https://boards.straightdope.com/u/Nen)\ **Post date:** [April 24, 2001, 5:31am UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/5 "2001-04-24T05:31:21Z")
Well, I’ve been on hiatus of late and after having spoken with **Gaudere** the other night, I felt I ought to return to her domain once again. So… > [@](#): > > \*Originally posted by Math Geek \* > That’s a tricky question to answer until you distinguish between math as “mathematical properties of the real world” and math as “formal systems of mathematics”. The real world doesn’t posess any _mathematical properties_, it only has _physical properties_. Mathematics is a construct of human mental faculties (although I suppose it is possible that other species may have some simliar notion). This construct is utilized to understand our perceptions by processing the information and assimilating it into the construct. The earth has physical properties of magnetic field, mass, motion, etc., and the nature of these properties is expressed through a formal system. Motion is not a mathematical property, but velocity can be quantified in terms of other constructs, namely space and time. Space and time are constructs in and of themselves, but they are also understood in terms of formal systems. > [@](#): > > \*Originally posted by Math Geek \* > As an example of the former, consider the phenomenon of “one”, i.e. unity. We certainly didn’t invent unity. But on the other hand we did invent the formal system known as arithmetic. Although I find your definition of the numeral one intriguing, I think it is misleading. Is the number one of mathematics really equivalent to the concept of unity? If I think of the universe as a whole, the concept of one and unity are synonymous. If I think of a number line, the concept of one and unity are not synonymous. Then again, I think unity is a concept, or construct, just as is the number one. > [@](#): > > \*Originally posted by Math Geek \* > Specifically, (if I may over-simplify for a moment here) the mathematician Peano set down the axioms and rules of arithmetic. Those axioms and rules together make up a formal system which _appears to describe_ certain natural phenomena like “one”, “zero”, and “addition”, but the formal system is not the same thing as the natural phenomena. Would you provide some examples of how the items you listed are phenomena in nature? I fail to understand how zero is an occurence in nature or how addition is a phenomenon. > [@](#): > > \*Originally posted by Math Geek \* > In other words, you and your friends are each partly correct: we invented a language to describe things that we discovered. In the same way we invented the colour wheel as one way of describing colours, and Newtonian mechanics as one way of describing the motion of the planets. That is precisely my point: we invent mathematical constructs to understand and express information about a physical world. However, I infer from your post that you don’t distinguish between mathematical and physical properties. > [@](#): > > \*Originally posted by ITR champion \* > Take the specific example of the 90 degree angle. Right angles have a lot of important occurences in “the real world”, so I doubt that any mathematical system could exist without giving them special signifigance. In force fields such as gravity or magnetic fields, you do no work moving a particle in the field if and only if you move the particle at right angles to the field. Right angles are significant in electromagnetism. Pi is significant in Euclidean geometry. _i_ is signficant in electrical engineering. Consider the equation (which I personally find to be blissfully aesthetic): e[sup]i\*pi [/sup]= -1 It reveals some wonderful mathematical properties…in the decimal system. If one tries to express that equation in binary, it ceases to have the same aesthetic. Any number can be quite significant pertaining to the right situation in the right formal system, but I doubt that makes mathematics predetermined. > [@](#): > > \*Originally posted by ITR champion \* > Although the development of mathematics could have gone slightly differently, the general principles are set and can’t be altered. Mathematics is continually being developed. At one time there was only Euclidean geometry, yet now there is Riemann geometry. At one time there wasn’t calculus. We continually _invent_ constructs to assist us in understanding our world around us, much as we continually create words to describe things outside our lexicon. > [@](#): > > \*Originally posted by Psi Cop \* > …if we invented mathematics, did we also invent physics? Or just invent the way to interpret physics? I’m sorry to save this question for last, but I thought that after my little diatribe it would be clear that I am inclined toward the latter. ---
**Author:** ![Nen](https://avatars.discourse-cdn.com/v4/letter/n/c5a1d2/32.png) [@Nen](https://boards.straightdope.com/u/Nen)\ **Post date:** [April 24, 2001, 5:47am UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/6 "2001-04-24T05:47:38Z")
> [@](#): > > \*Originally posted by Alessan \* > Plus, if you put a wall up at any angle other than 90 degrees from the ground, it’ll fall over. Oh yeah, two more things: 1) I don’t like you dissin’ my Inuit friends and their igloos, and 2) I used to be a carpenter–their definition of a right angle tends to be a bit less rigid than that of a mathematician. 😃 ---
**Author:** ![SPOOFE](https://avatars.discourse-cdn.com/v4/letter/s/c5a1d2/32.png) [@SPOOFE](https://boards.straightdope.com/u/SPOOFE)\ **Post date:** [April 24, 2001, 7:22am UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/7 "2001-04-24T07:22:13Z")
Humans only invented the terms and definitions that make mathematics understandable. The same way they make a distinction between “granite” and “limestone”. Math is certainly a “natural” force - sort of - in that you can change the terms or definitions, but “1” will still be “1”, “2” would still be “2”, and “69” would still be “69”. An integer by any other name would smell as sweet. ---
**Author:** ![Cabbage](https://avatars.discourse-cdn.com/v4/letter/c/f07891/32.png) [@Cabbage](https://boards.straightdope.com/u/Cabbage)\ **Post date:** [April 24, 2001, 10:22am UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/8 "2001-04-24T10:22:38Z")
> [@](#): > > Consider the equation (which I personally find to be blissfully aesthetic): > > e[sup]i\*pi[/sup] = -1 > > It reveals some wonderful mathematical properties…in the decimal system. If one tries to express that equation in binary, it ceases to have the same aesthetic. Decimal vs. binary has nothing to do with it, the equation is precisely the same regardless of what numerical base you’re using. As for the OP, consider some of the most elementary levels of math. Suppose there are other civilizations of intelligent beings in the universe. I find it incredibly difficult to believe that they would not also develop the notion of _counting_ things, just as we have done. The language, notation, and so forth used would most definitely be different, but the ideas involved would necessarily be the same. Perhaps counting itself is an invention used to describe properties of the universe, but when that invention is invariant throughout the universe, I’m more inclined to describe it as a _discovery_. From there, things such as addition and multiplication would be immediate (If I have _n_ things and he has _m_ things, how many things do we have together? If _n_ of us each has _m_ things, how many things do we have altogether?). The idea of prime numbers would not be far behind (Can I or can I not divide these objects up evenly among everyone?), and all of these notions would necessarily be identical to our own. Thinking of it in that sense, what room is there for “invention”? ---
**Author:** ![december](https://avatars.discourse-cdn.com/v4/letter/d/838e76/32.png) [@december](https://boards.straightdope.com/u/december)\ **Post date:** [April 24, 2001, 12:54pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/9 "2001-04-24T12:54:27Z")
One of the Socratic dialogues involves a discussion of whether we are BORN knowing mathematics. Socrates argues that mathematics training is really helping the student to remember what he already knoew. This position is not widely held today (to say the least). One way to more precisely define the OP would be to ask whether a race of intelligent aliens would be expected to devleop the same mathematics. In fact, the Search for Extra-Terrestrial Intelligence is beaming some mathematcs into space. I think SETI is using something like the values of _pi_ or _e_ in base 2. Their approach supports the idea that some mathematics is natural or discovered. However, having spent a long time as a math grad student, I believe that much mathematics is totally invented, and for a specific purpose – to get an article published. ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 24, 2001, 3:18pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/10 "2001-04-24T15:18:20Z")
Mathematics is a language (several languages, actually). The fact that the languages of math can usefully describe aspects of the world does not mean that the particular syntax invoved was “discovered”. Does the word “rock” mean that English was partly discovered? Mathematics does not exist outside of the human mind. Much as I admire the Socratic Dialogues, mathematics also does not exist as an intrinsic property of the human mind. Thus, it cannot be discovered. > [@](#): > > _re: SETI_ > Their approach supports the idea that some mathematics is natural or discovered. No. It supports the idea that any intelligent life advanced enough to allow interstellar communication would most likely have invented tools for describing the physical properties of the Universe. Binary mathematics is one such tool, and the extremely simple nature of its syntax makes it a good candidate for broad understanding. ---
**Author:** ![Orbifold](https://avatars.discourse-cdn.com/v4/letter/o/779978/32.png) [@Orbifold](https://boards.straightdope.com/u/Orbifold)\ **Post date:** [April 24, 2001, 4:15pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/11 "2001-04-24T16:15:12Z")
> [@](#): > > \*Originally posted by Nen \* > \*\* > > > [@](#): > > > > \*Originally posted by Math Geek \* > > That’s a tricky question to answer until you distinguish between math as “mathematical properties of the real world” and math as “formal systems of mathematics”. > > The real world doesn’t posess any _mathematical properties_, it only has _physical properties_.\*\* What do the following things have in common: a pair of apples, a pair of motorcycles, a pair of hats, and a pair of birds? I’m sure there’s more than one answer, but one answer is that there’s a pair of each. _That_ is what I call a “mathematical” property of the real world, the fact that some sets of objects share the same cardinality. The number “two” is the word we use to describe the property of having the same cardinality as a pair of apples. And as **Cabbage** said, once you recognize things like cardinalities and ordinalities, other mathematical properties of the real world like addition start to become apparent. (Perhaps you recognize those properties but refer to them as “physical” ones. In which case we’re just arguing over a definition…) > [@](#): > > \*\* > > > [@](#): > > > > \*Originally posted by Math Geek \* > > As an example of the former, consider the phenomenon of “one”, i.e. unity. We certainly didn’t invent unity. But on the other hand we did invent the formal system known as arithmetic. > > Although I find your definition of the numeral one intriguing, I think it is misleading. Is the number one of mathematics really equivalent to the concept of unity?\*\* Nope. That’s why I put “one” in quotes. The number one, to me, belongs in the world of formal mathematical systems, whereas the phenomenon of being only one thing (instead of two things, or no things) is a observable feature of reality. Originally the ancient Greeks (possibly the first to think about the philosophy of mathematics) thought about numbers this way: as a way of describing certain observed properties of things around them. Specifically, some things are numerous and others aren’t. (The Greek philosophy of numbers goes farther than that, of course: Platonic forms and all that. But I’m not going there.) It’s only in modern times that arithmetic has moved away from being a “physics of counting” to an exploration of a formal system. But the dichotomy between the formal system and the real world is still there. The difference between the two was cast into very stark relief by Godel’s Incompleteness Theorem, which among other things draws a sharp distinction between the _truth_ of a number-theoretical statement (i.e. how well it models the behaviour of the real world) and its _provability_ (whether or not it follows from the axioms of the formal system). > [@](#): > > **Then again, I think unity is a concept, or construct, just as is the number one.** That summarizes our philosophical differences in a nutshell…or almost: > [@](#): > > **However, I infer from your post that you don’t distinguish between mathematical and physical properties.** I believe that the universe has both physical and mathematical properties. (Although I’d be willing to bet that those categories overlap here and there.) > [@](#): > > \*Originally posted by Cabbage \* > **From there, things such as addition and multiplication would be immediate (If I have n things and he has m things, how many things do we have together? If n of us each has m things, how many things do we have altogether?). The idea of prime numbers would not be far behind (Can I or can I not divide these objects up evenly among everyone?), and all of these notions would necessarily be identical to our own. Thinking of it in that sense, what room is there for “invention”?** There are other systems of “arithmetic” out there. It’s just that the ones that differ from conventional number theory either don’t describe the real world very well (“One, two, three, many.”), or else they’re extensions of conventional number theory like the ones generated by Godel’s theorem, which differ only in some bizarre axioms which may or may not reflect real-world behaviour. Its true that conventional number theory seems to be _the_ theory to describe counting. We certainly don’t _expect_ to make some new observation about the real world which forces us to update the theory, in the way that the orbit of Mercury forced us to improve upon Kepler’s and Newton’s laws. But perhaps somewhere some alien civilization (presumably with a reeeeally long lifespan) has sat down and counted some really big piles of frobniks, and discovered something which will someday force us to add a new Godel-like axiom to number theory. Okay, so maybe even I don’t believe that. But it’s theoretically possible 😃 ---
**Author:** ![Tymp](https://avatars.discourse-cdn.com/v4/letter/t/c5a1d2/32.png) [@Tymp](https://boards.straightdope.com/u/Tymp)\ **Post date:** [April 24, 2001, 5:04pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/12 "2001-04-24T17:04:22Z")
> [@](#): > > \*Originally posted by Math Geek \* > **What do the following things have in common: a pair of apples, a pair of motorcycles, a pair of hats, and a pair of birds?** These things are not pairs until I decide to see them as such. Until then, I see only apple, apple, motorcycle, motorcycle, hat, hat, bird, bird. Only when I seek to define and describe relationships between these things do I fire up the math. The relationships between physical things described with mathematics only exist when I employ mathematics to describe them. I could just as easily describe them as red, black, gray, yellow. They are only pairs when I need them to be. From this, I believe that mathematics, like poetry and song, is a language invented for the specific purpose of describing the observed world in a way made useful only through the developed understanding of that language. Of course, I’m a bit out of my element here, and you are all more than welcome to point and laugh. ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 24, 2001, 6:11pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/13 "2001-04-24T18:11:55Z")
~points at **Tymp** and laughs~ But only because you nailed it. Even stuck the dismount. **10** ---
**Author:** ![Arnold\_Winkelried](https://avatars.discourse-cdn.com/v4/letter/a/3d9bf3/32.png) [@Arnold\_Winkelried](https://boards.straightdope.com/u/Arnold_Winkelried)\ **Post date:** [April 24, 2001, 6:31pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/14 "2001-04-24T18:31:20Z")
I agree with MathGeek: “we invented a language to describe things that we discovered.” e.g. I am writing a computer program using the most efficient algorithms to find digits of pi. Am I “inventing” the digits of pi or “discovering” the digits of pi? I say that I am “discovering” them, since these are values that exist independently of the method I use for my calculation (and even exist in the absence of anyone attempting that calculation). When I prove a theorem that states that there is no largest prime number, I am demonstrating a fact that already existed even before the first human had imagined the “concept” of a prime number. How can one say that I am “inventing” this statement? I am “inventing” the proof, but I am “discovering” the fact that there is no largest prime number. ---
**Author:** ![Nen](https://avatars.discourse-cdn.com/v4/letter/n/c5a1d2/32.png) [@Nen](https://boards.straightdope.com/u/Nen)\ **Post date:** [April 24, 2001, 6:44pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/15 "2001-04-24T18:44:36Z")
**Spiritus Mundi** and **Tymp** , I wholeheartedly concur. But since you asked, **Tymp** , I must now point and laugh at you. 😉 > [@](#): > > _Originally posted by SPOOFE Bo Diddly_ > Humans only invented the terms and definitions that make mathematics understandable. The same way they make a distinction between “granite” and “limestone”. But mathematics is not an entity in itself. As **Spiritus Mundi** , **Tymp** and I have attempted to point out in various ways, mathematics is a language or construct. Granite is a mineral consisting of such-and-such elements in such-and-such lattice. Granite exists physically independent of our attribution of nomenclature. The rock itself does not inherently possess the nature of the label “granite.” Nor do two rocks inherently possess the labels of “one” and “two.” Concepts of “oneness,” “twoness,” and “graniteness” are attributes we apply to objects—they are not attributes of the object itself. There exists a significant distinction between naming something and the actual properties of that something. > [@](#): > > \*Originally posted by Cabbage \* > Decimal vs. binary has nothing to do with it, the equation is precisely the same regardless of what numerical base you’re using. I concur, but you’re missing my point. This appeal to discovery seems to be partially based on aesthetics. Referring to the numerous examples of right angles in nature does not make right angles inherent properties of an object. It is our perceptions, and the constructs through which we describe our perception, which possess this aesthetic. My point was that that particular equation is aesthetic only in a particular system or language. When one changes the language, the aesthetic changes. > [@](#): > > \*Originally posted by Cabbage \* > Suppose there are other civilizations of intelligent beings in the universe. I find it incredibly difficult to believe that they would not also develop the notion of counting things, just as we have done. The language, notation, and so forth used would most definitely be different, but the ideas involved would necessarily be the same. Perhaps counting itself is an invention used to describe properties of the universe, but when that invention is invariant throughout the universe, I’m more inclined to describe it as a discovery. > > From there, things such as addition and multiplication would be immediate (If I have n things and he has m things, how many things do we have together? If n of us each has m things, how many things do we have altogether?). The idea of prime numbers would not be far behind (Can I or can I not divide these objects up evenly among everyone?), and all of these notions would necessarily be identical to our own. Thinking of it in that sense, what room is there for “invention”? Again, mathematics is a language. One (human or alien) may enumerate, add or divide objects. Those actions are descriptions of our perceptions. Stating that the formal systems of numbers and arithmetic are discoveries is like stating that the formal systems of phonetics and syntax are discoveries. The capacity to wield language is based on physiology. The capacity to understand language is based on physiology. The innate-ness of language in the environment is absurd. As I stated above, “Concepts of ‘oneness,’ ‘twoness,’ and ‘graniteness’ are attributes we apply to objects—they are not attributes of the object itself.” Mathematical properties are not properties of the physical world—they are properties of the construct we employ to understand the physical world. > [@](#): > > _Originally posted by Math Geek_ > What do the following things have in common: a pair of apples, a pair of motorcycles, a pair of hats, and a pair of birds? I’m sure there’s more than one answer, but one answer is that there’s a pair of each. That is what I call a “mathematical” property of the real world, the fact that some sets of objects share the same cardinality. The number “two” is the word we use to describe the property of having the same cardinality as a pair of apples. Yet that cardinality of “pair-ness” is a construct. It is a result of our perception; it is not inherent in the objects themselves. > [@](#): > > \*Originally posted by Cabbage \* > (Perhaps you recognize those properties but refer to them as “physical” ones. In which case we’re just arguing over a definition…) It is precisely that definition which is at the root of this debate. Coming to an acceptable definition is essential to arriving at a resolution. ---
**Author:** ![Nen](https://avatars.discourse-cdn.com/v4/letter/n/c5a1d2/32.png) [@Nen](https://boards.straightdope.com/u/Nen)\ **Post date:** [April 24, 2001, 6:59pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/16 "2001-04-24T18:59:13Z")
Dammit, **Arnold** , you snuck this one in on me. > [@](#): > > \*Originally posted by Arnold Winkelried \* > I agree with MathGeek: “we invented a language to describe things that we discovered.” I concur as well, but… > [@](#): > > \*Originally posted by Arnold Winkelried \* > When I prove a theorem that states that there is no largest prime number, I am demonstrating a fact that already existed even before the first human had imagined the “concept” of a prime number. How can one say that I am “inventing” this statement? I am “inventing” the proof, but I am “discovering” the fact that there is no largest prime number. How does a number, an element of a language, exist prior to the conception of the language? In discovering the digits of pi or discovering that there is no largest prime number, you are discovering properties of the language which has been invented. You may be discovering properties of the language which has been invented (much as linguistics is an analysis of the properties of languages), but those properties aren’t inherent physical properties of the world–they are properties of a construct. Constructs, models, languages, et cetera aren’t discovered. {fixed code. --Gaudere} [Edited by Gaudere on 04-24-2001 at 08:06 PM] ---
**Author:** ![FriendRob](https://avatars.discourse-cdn.com/v4/letter/f/ecc23a/32.png) [@FriendRob](https://boards.straightdope.com/u/FriendRob)\ **Post date:** [April 24, 2001, 7:06pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/17 "2001-04-24T19:06:23Z")
Yes, _invented_ - even proofs in number theory are only true _within_ a particular mathematical system. I used to subscribe to the idea that mathematics existed “out there” and all we did was discover its eternal truths. Then I read _The Mathematical Experience_ by Davis and Hersch. You can get different geometries by affirming or denying the “parallel postulate”. You can get different ideas about the real numbers by affirming or denying the “continuum hypothesis”. Then you look at the world and ask which of these systems can be mapped onto it so that the correspondence is meaningful. Goedel’s theorem tells us that _any_ mathematical system has undecidable propositions- this includes systems that deal only with integers. In other words, there is some statement like “All prime numbers have the property that …” which is true for one definition of integers and false for some other definition, but where _both_ definitions include the Peano postulates. ---
**Author:** ![Spiritus\_Mundi](https://avatars.discourse-cdn.com/v4/letter/s/c68b51/32.png) [@Spiritus\_Mundi](https://boards.straightdope.com/u/Spiritus_Mundi)\ **Post date:** [April 24, 2001, 7:17pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/18 "2001-04-24T19:17:11Z")
But Arnold, those are statements that we make **with** mathematics. We invent the ships that allows us to discover the new lands. “No largest prime” is not mathematics. This sentence is not the english language. ---
**Author:** ![Arnold\_Winkelried](https://avatars.discourse-cdn.com/v4/letter/a/3d9bf3/32.png) [@Arnold\_Winkelried](https://boards.straightdope.com/u/Arnold_Winkelried)\ **Post date:** [April 24, 2001, 8:41pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/19 "2001-04-24T20:41:26Z")
Nen: _How does a number, an element of a language, exist prior to the conception of the language? In discovering the digits of pi or discovering that there is no largest prime number, you are discovering properties of the language which has been invented._ But the digits of pi, or the fact that there is no largest prime number, is independent of the properties of the language I am using. (Disregarding of course the semantic differences between pi expressed in base 10 and pi expressed in base 16). If I tie a string to a stake planted in the sand, stretch the string as far as it can go and draw a line, I end up with a circle. The ratio between the line drawn in the sand and the length in the string will exist whether or not I invent a language to describe it. FriendRob: _Goedel’s theorem tells us that any mathematical system has undecidable propositions- this includes systems that deal only with integers. In other words, there is some statement like “All prime numbers have the property that …” which is true for one definition of integers and false for some other definition, but where both definitions include the Peano postulates._ But some statements are true regardless of your definition of the integers. Or are you saying that one could come up with a definition of the integers in which there would be a largest prime number? I thought Goedel’s theorem said that some statements will be undecidable, but it did **not** follow that “for any statement X, one can invent a system in which statement X will be demonstrably false.” Spiritus Mundi, we seem to have a different definition of mathematics. I use the term mathematics to include the methods or language we use to find facts about (e.g.) the natural numbers, but also the facts themselves, whereas you restrict mathematics to the first clause of my definition. If I say “my observations show that there is a nova at position X in the sky”, what would be the proper relationship of that sentence with the word “astronomy”. Would you say “that is a fact that I proved using the methods of astronomy”? i.e. astronomy describes the constructs I use to find out facts about the universe? ---
**Author:** ![Nen](https://avatars.discourse-cdn.com/v4/letter/n/c5a1d2/32.png) [@Nen](https://boards.straightdope.com/u/Nen)\ **Post date:** [April 24, 2001, 9:14pm UTC](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084/20 "2001-04-24T21:14:24Z")
> [@](#): > > \*Originally posted by Arnold Winkelried \* > But the digits of pi, or the fact that there is no largest prime number, is independent of the properties of the language I am using. That’s not the case. If one constructs a language to enumerate objects, the elements of that language, i.e., numbers, are not independent of the properties of the language. Consider the English language. The term “tree” is a name applied to vegetation meeting specific requirements. A linguistic analysis shows that the term in question is a noun. One can speak English without knowing what a noun is, much as one can be unaware of all of the digits of pi, but the elements of a language are defined by the language–they are not independent of it. > [@](#): > > \*Originally posted by Arnold Winkelried \* > If I tie a string to a stake planted in the sand, stretch the string as far as it can go and draw a line, I end up with a circle. The ratio between the line drawn in the sand and the length in the string will exist whether or not I invent a language to describe it. Again, I disagree. The ratio of which you speak is not physical property. Physical properties of the scenario you describe displacement and mass. To describe how it appears one applies the term “circle.” The name “circle” does not exist without language, nor does the term “ratio” or it’s value. These terms are elements of the construct. An element of the construct cannot exist without the invention of the construct. If you disagree, can show me a ratio without language, mathematical or otherwise? [Next page](https://boards.straightdope.com/t/mathematics-invented-or-discovered/63084.md?page=2)