Afternoon Inquisition

AI: The Mathman Prophecies

I heard an interesting discussion recently, and thought it would make for a good Afternoon Inquisition. It was thought provoking and engaging.

Of course, it’s possible no one here will find it as interesting as I did, in which case I’ll look pretty silly for posting it. But do me a favor. Even if this is the most boring question you’ve ever heard, throw me a bone and discuss the shit out of this sumbitch. Let’s push for a thousand comments!

The question is simple:

Was mathematics discovered or invented?

The Afternoon Inquisition (or AI) is a question posed to you, the Skepchick community. Look for it to appear daily at 3pm ET.

Sam Ogden

Sam Ogden is a writer, beach bum, and songwriter living in Houston, Texas, but he may be found scratching himself at many points across the globe. Follow him on Twitter @SamOgden

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  1. @SkepLit:


    Srsly tho, I vote for “discovered”. When we make contact with extraterrestrials (should be sometime next Tuesday), they’ll probably have vastly different symbology for the whole deal, but likely expressing the same concepts. And if any of it differs, it’ll likely be stuff we (or they) didn’t think of yet but would’ve worked out eventually.

  2. John P. CroMagnon did not flip over a log and exclaim, “Look! Mathmatics!” At some point our ancestors noticed a relationship in numbers and invented a language, mathmatics, to describe the relationship.
    Mathmatics was no more “discovered” than any other human language.

  3. Oh boy. Sam, I think you just earned Wikipedia a few hundred extra hits it might not have otherwise gotten.
    Well, I would think that math has sort of evolved along with us as we’ve used it. Probably starting with mundane counting, through simple geometry and then passing on into the complex theoretical musing done today. I never really given it much thought but that’s my ‘off the top o’ my head’ answer.

  4. One of my favorite quotes from my 1st year Physics prof: “God did not make the X-Y Plane. A clever mathematician made the X-Y plane.”

    The laws of the universe are already there, but they are not defined in terms of the math used to describe them. Mathematics is a human language created to express and understand them as they are discovered.

  5. @revmatty:

    I like to think of most science as simply an approximation of nature. I agree that we invented the symbology, but I can’t help but think we invented some of the other aspects of math, too. We use perfect circles and right triangles and exact degrees and many other mathematical tools that do not actually exist in the real world, right?

  6. I don’t really think you can ever discover a concept because your concept may be wrong.

    I’m going to go with invented all the way around. We dreamed up the concepts in math and the language to describe them. Take, for example, pi. We say that the ratio of a circle’s circumference to its diameter is a transcendental number, but is it? Maybe we’ll run into an alien species that just laughs at us and tell us we’re not thinking about it right. I don’t think you can say we discovered pi until you can prove it exists as anything more than a human invention. As are circles. They’re just a special case of an ellipse, and an ellipse is just a special case of a conic section. Would an alien species that only dealt with conic sections in the abstract have failed to discover the circle or just have a different way of thinking about math?

  7. Mathematics was discovered in the same way that literature is discovered: since every possible permutation of letters exists, all one does is uncover a particular arrangement that becomes “great literature.”

  8. I’d agree that mathematical concepts such as notation and ways to manipulate numbers were–and are being–invented. But I think that other aspects of mathematics–e.g. at least some ratios & constants–are basic to our universe and so were discovered. So I’d have to say mathematics writ large was both.

  9. Look at all these people, being all sensible and defining their terms. Next thing you’ll tell me is an unobserved tree falling makes a sound if we define “sound” as vibrations in the air, but not if we define it as the sensation produced when those vibrations reach an observer’s ear.

    What kind of bullshit armchair philosophy session is this?!

  10. I think mathematics was invented.. The most recent Scientific American touches on this a little, and sums it up best, I think. It’s a tool.. we made it.

    There are phenomenon that require math for our understanding.. we use this tool to understand the bigger part.

    And, to be sure, it’s a great tool, because it builds upon itself in ways most human invention doesn’t get a chance to!

  11. @dougreardon: since every possible permutation of letters exists, all one does is uncover a particular arrangement that becomes “great literature.”


    Maybe. But any race that computes will use computers that have the same theoretical limits. The halting problem isn’t subjectively judged as “great”, it’s a definable and absolute property of an abstract object, and that abstract object will have the same exact properties no matter who discovers it or when.

    Our models of the universe are approximations. This is true. But prime numbers will exist in any formal language that is powerful enough to perform basic arithmetic. They are a property of a class of formal systems that was discovered, not invented.

  12. If a mathematical formula is true to itself then it seems to me it was perhaps more of a discovery than an invention. I’m also wondering if any particular philosophical schools of thought have addressed this question. Given the known chemical/electrical/biological properties of humans it seems to me there is a preexistent possibility for any and all known or knowable mathematical formulas. That implies discovery over invention in my mind.

  13. It was discovered. I remeber one day about 65 million years ago I was running like hell while a herd of wooly petrodactyls were chasing me. I tripped over a wild quadratic equation and fell into an imaginary number. Well I stayed hidden until the last pursuer had slithered away. When I finally stuck my head up to look around I was staring straight into the golden mean. I tried to capture it but it was to fast. I tracked it for days and eventually it led me to the hidden valley of mathmatics. I’ve been working the number mines ever since. And it has made me wealthy beyond irrational dreams.

  14. @James Fox: If a mathematical formula is true to itself then it seems to me it was perhaps more of a discovery than an invention.

    A formula is only true once you have accepted the required axioms. Different axioms lead to different math. Euclidean and non-Euclidean geometry are both “true”. They just have different starting points. I like Euclidean geometry because it helps me build furniture. I’d like any other mathematical system that accomplished the same goal, but I don’t think humans discovered Euclidean geometry any more than we discovered the table saw.

  15. Science is a process of discovering things about our universe, but mathematics is bigger than that. Euler’s identity is true in every universe there could possibly be. Powerful stuff.

    So we invent ways of “doing” mathematics, and we can come up with novel ways of using its findings, but the Truth is there to be discovered.

  16. @davew: Euclidean and non-Euclidean geometry are both “true”.


    And Gödel incompleteness applies to both of them… and to every other formal system that can accomplish a certain kind of task. Every possible system created by god, man, or aliens that can calculate an orbit and a velocity is incomplete in exactly the same way.

    Incompleteness is a discovery that concerns inventions.

  17. We didn’t invent circles – they’re just what happens when you swing something around a fixed point. They’re the shape that forms when you drop a pebble straight into water. These are not inventions.

  18. In the same way that elements such as Helium existed before they were ‘discovered’, mathematics is an integral part of the universe itself.
    The methods and processes used to extract or purify these elements may have been invented, but the elements themselves existed before people learned how to use them.
    This analogy can be stretched (probably a little too far) to the extremes where chemistry becomes particle physics, then quantum physics which in turn ends up being the realm of pure math.

  19. As a mathematician, I’m going to totally cop out.

    Axioms were invented. Everything past that was discovered. But, it happened backwards… we discovered theorems, then back-engineered to our axioms – which it turns out we had invented in the first place in order to discover theorems.

  20. @jtradke:

    We didn’t invent circles – they’re just what happens when you swing something around a fixed point. They’re the shape that forms when you drop a pebble straight into water. These are not inventions.

    Just to play devil’s advocate here: we can describe a perfect circle mathematically, but nowhere is there such a thing in nature that makes any sort of sense. Surely that’s an invention of ours that exists only in our imaginations.

    Or how about finding the volume of a, say, 40-dimensional object? I sure had my fair share of this sort of thing in my college calculus class.

  21. Part of me hopes that the title of tonight’s AI is a reference to Mathman, the number munching hero from Square One TV (a math centered kids show that ran on PBS in the early 90’s). I used to watch that damn show all the time when I was little. Ah, memories…

  22. @davew:

    You need math to count boobies and bacon.
    I’m going with largely discovered the principals and inventing terms and concepts.

    There’s one tit, and another tit, count them up that’s two! Two boobies ha ha ha. A lovely pair of breast.

  23. @Aaron:

    Look at all these people, being all sensible and defining their terms. Next thing you’ll tell me is an unobserved tree falling makes a sound if we define “sound” as vibrations in the air, but not if we define it as the sensation produced when those vibrations reach an observer’s ear.

    What kind of bullshit armchair philosophy session is this?!

    If we approach the question of the tree falling in the woods from a quantum perspective, then the tree both makes a sound, and does not make a sound. It exists in both a sound-making and non-sound making state simultaneously The result we don’t know until the event is observed.

    However, we can express the unknown result as a quantum wave function, representing the probability of the tree either making a sound, or not making a sound.

    Now we know that when the falling of the tree is observed, that the tree will make a sound 100% of the time, assuming extrordinary circumstances, such as the tree falling in a vacuum, or other scenarios not considered in this example. So when the quantum wave function collapses, we know that the tree will make a sound. Therefore, it is reasonable to surmise that in the absence of an observer the probability of a tree not making a sound is 0.

    Q.E.D., Bitchez!

  24. Mathematics is the only completely man-made science: every single bit of it was created out of nothing. (Heh!) The fact that it all hangs together and is not self-contradictory simply shows that each bit, added on to the previously-created mass, was only acceptable as part of the mass if it didn’t contradict what went before.

    The better question is whether math is a science or an art.

  25. As may have been previously brought up, we do invent the axioms to derive mathematical statements. Even those are flawed (see Godel’s incompleteness theorem) though. In the end though, the world still insists on being a nice representation of math.

    So as a mathematical physicist, I say the answer to that question is definitely not known — there are camps even within math and physics and no consensus. Really the fact that we ask the question is the most interesting part of it. Personally though, I think it is “useful” to think we invented mathematics. Throughout history we have discarded certain things in math because we thought math was part of the universe and so we didn’t like the new — what we viewed as contrived — creations. Look at irrationals (the Greek’s hated them), negative numbers, complex numbers, and more. Thinking math is part of nature has held us back in some ways. Though thinking math is part of nature has also spurred much research as well.

  26. Is it perhaps a question of semantics: does ‘mathematics’ refer to the notation to describe a system, or the system itself?

    I say the former was invented; the latter, being properties of the system, would suggest discovery.

    Where you draw the line remains blurred, however; I guess that’s more the point of this AI. :-)

    Mark C. Chu-Carroll (Good Math, Bad Math) had a good post on how most numbers cannot be written down in any form — to me this suggests that it may have some independent existence, but I couldn’t qualify it as any more than a feeling.

  27. It seems to me that this is a bit like the “if a tree falls in the woods …” question. The trick comes from the vagueness of our language, not from the question.

    If by mathematics you mean the fundamental rules governing sets and numbers then they were discovered. These relationships are basic logic, they are true by necessity and therefore are true irrespective of whether or not an intelligent mind has ever considered them.

    If you mean the notations and language we have to describe these things, then they were created.

  28. @Chakolate:

    The question of whether Math is a science or an Art almost made me fall out of my chair. Not in a good way, either.
    Your phone and MP3 player won’t suddenly stop working because the Fast Fourier Transform has suddenly gone out of fashion.
    Art is all about vague terms like beauty and form. Maths however is solid and closed to interpretation.

    You don’t get to hear about mathematics papers where somebody faked or forged the results, (except in regard to statistics).
    It exists and it has always existed, the only part the we as humans have to play in this process is to discover our way through this maze and find ways to use these discoveries .

  29. Math does describe the world in a factual way. It also describes worlds that may/may not exist, too. Consider multiple-space geometries beyond our three-dimensional one, for example. IMHO, that alone makes it worthy of being called a “discovered science.” Sort of like RevMatty says. It was always inherent in our universe, but we discovered how to use it to describe the universe around us.

    My “confession” about my lack of math skills above was to explain that I’m not terribly qualified to judge and to take my opinion as just that – a barely informed one.

  30. Math is just an expression of what is already there. You might as well ask: Did we invent color? If you’re asking where the word “orange” came from, that is an invention, but the underlying physics that involve color absorption and reflection were always there.

    Similarly, Math is human construct, but the values and how they interact have always been there. An alien race might call 2+2=4 “jerb+jerb=raf” and might call what they’re doing “fwanga” but that doesn’t change the fact that they and we are describing exactly the same thing.

    And actually if you think about it, color is really more created than discovered. Think about it, if a human and an alien were looking at an apple, the color absorption of their eyes might be different enough to make an apple look more like an orange to them. Then it really would make sense to compare apples and oranges :)

  31. I am a little late to this discussion but what the heck…

    When one looks at mathematics as a language, then one has to go with “It was invented.” Languages aren’t discovered. Once the language was invented many principles of mathematics were discovered. The principles of mathematics have always been there and had to be discovered however this couldn’t be done until a language was invented that allowed human beings to talk about those principles.

    If that makes any sense at all.

  32. Interesting question even though it’s not my best subject. I really have no clue. It is important to have at least basic math skills though. We need to know how much money we have at the time or basic fractions to measure ingredients when we cook. Math is all around us and my teacher was right when she said it is one of the most useful things we learn in school. I think that it is possible that it may have been discovered.

  33. Aeon65: “When one looks at mathematics as a language,”

    That’s the problem. It isn’t strictly a language. There are a lot of subjective terms in a language. Some languages don’t have the necessary structure for some things, and when they do, some things mean different things to different cultures. Mathematics isn’t like that. The value of 2 might be called by many different names, but it means EXACTLY the same thing in all languages, and adding 2 “twos” in any language will give you the same result, even though it is also called by many different names. Our EXPRESSION of mathematics is part of our language, and was invented, but mathematics itself is universal and merely discovered.

  34. rasmur: “Mathematics is not reality but a model of reality. Therefore, mathematics was invented and based on the nature of reality.”

    “Reality is not real, simply our interpretation of the universe. Therefore, reality is invented and based on the nature of the universe.”

    “Evolution is not real, simply our expression of the function of biology. Therefore, evolution is invented and based on the nature of biology.”

    I think you’re slicing the bread too thin.

  35. @Amy:
    Though I don’t disagree with Sam+AI=Awesome (kudos there), I do question your choice of Language. “Basic Math was discovered”. Where do you stop? Addition and subtraction? Multiplication and Division? Powers and Roots?
    Calculus may be out of the range of your average bear, but even little Johnny No-Stars can still understand Miles per hour and 0-60 in 4.5 Seconds. So at what point does maths stop being discovered and become invented?
    Fermat’s Last Theorem, [sorry for dumping here] No three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than two; was proved in ’94.
    Was it any less true in 1884? 1994BCE? I know it would be unlikely that I could ever hope to fully understand the proof, but I won’t deny that it existed before the theorem or the proof were even conceived.
    So can anybody point out Maths that was ‘invented’?
    I admit I will believe that Andrew Wiles et al. proved the theorem, without my being able to check the proof myself. But I do believe that other people have the necessary background to do the checking for me…
    …and now I stumble clumsily back to my point. Maths is not open to interpretation, (aside from it’s bastard sibling, Statistics) but exists independent of the minds that conceive the pathways of Discovery though it’s convoluted ummm… pathways.
    Language can fail, flair or falsify. Mathematics simply is.

  36. Math was invented in the Dark Ages right after the invention of the Holy Hand Grenade of Antiock…to keep the monks from blowing themselves to smithereens. You see, the counting must be three…

  37. @aeon65: When one looks at mathematics as a language, then one has to go with “It was invented.” Languages aren’t discovered.


    When you say “language” do you mean “notation?” A language is a very strict mathematical concept, and I would argue that the properties of formal languages are discoveries.

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