Did we discover mathematics or invent it?

September 20, 2022 • 1:30 pm

Here’s an op-ed in the NYT about mathematical realism (also called “mathematical Platonism”). Do numbers and math exist out there somewhere and are real entities, or are they the product of human contrivance? Did humans invent mathematics or discover it?

My own view, and I’m hardly qualified to express one as I’m not a philosopher, is that nature can be expressed in mathematical rules because nature (or at least physics) is regular. That is, there are laws of nature most prominently the laws of physics. Now I don’t know why that is, but if there are laws and regularities, then we can always express them in math. And for the life of me I can’t see that numbers predated minds that could comprehend them—that’s almost a religious view (note the title of Wilkinson’s book, though I have no idea whether he’s religious). And you’ll never convince me that the Pythagorean Theorem is somehow floating out there in the cosmos, and is not just a regularity noticed and expressed in mathematical form by humans.

But on this I have little expertise.  Click to read.

A quote:

The beginner math mystery, available to anyone, concerns the origin of numbers. It’s a simple speculation: Where do numbers come from? No one knows. Were they invented by human beings? Hard to say. They appear to be embedded in the world in ways that we can’t completely comprehend. They began as measurements of quantities and grew into the means for the most precise expressions of the physical world — E = mc², for example.

The second mystery is that of prime numbers, those numbers such as 2, 3, 5, 7, 11 and 13 that can be divided cleanly only by one or by themselves. All numbers not prime are called composite numbers, and all composite numbers are the result of a unique arrangement of primes: 2 x 2 = 4. 2 x 3= 6. 2 x 2 x 2 = 8. 3 x 3= 9. 2 x 3 x 3 x 37 = 666. 29 x 31 = 899. 2 x 2 x 2 x 5 x 5 x 5 = 1,000. If human beings invented numbers and counting, then how is it that there are numbers such as primes that have attributes no one gave them? The grand and enfolding mystery is whether mathematics is created by human beings or exists independently of us in a territory adjacent to the actual world, the location of which no one can specify. Plato called it the non-spatiotemporal realm. It is the timeless nowhere that never has and never will exist anywhere but that nevertheless is.

Mathematics is one of the most efficient means of approaching the great secret, of considering what lies past all that we can see or presently imagine. Mathematics doesn’t describe the secret so much as it implies that there is one.

But is there a “great secret”?  I can’t imagine Wilkinson is thinking of God, for he doesn’t allude to a divinity. What else could a great secret be, though? To the mystery of prime numbers, my own response would be, “Well, that’s just the way it is.”

By the way, both the existence of regular physical laws and mathematics have been used as evidence for God: see  pp. 158-160 of Faith Versus Fact. 

73 thoughts on “Did we discover mathematics or invent it?

  1. Timely. I spent six hours on Sunday evening with a young mathematician (enrolled in a PhD) program at Boston College. She is working on something tangential to proving whether prime numbers are infinite. I will ask her.

    1. It’s been known for a great while that primes are infinite, and multiple proofs exist. What is the something tangential? (I have a special interest in palprimes).

        1. Cool! That means the set of emordniprimes must be infinite too. Now we have to wait for some poor soul to slog their way through a marathon proof just because you can’t find your book. Well done!

    2. Assume there is a largest prime P. Now take the product of all the numbers from 2 to P and add 1 to it. Call it N. i.e. N = 2 x 3 x 4 x 5 x …x (P – 1) x P + 1.

      N is not divisible by 2 because its remainder would be 1.
      N is not divisible by 3 because its remainder would be 1.
      N is not divisible by 4 because its remainder would be 1.
      N is not divisible by 5 because its remainder would be 1.

      N is not divisible by (P – 1) because its remainder would be 1.
      N is not divisible by P because its remainder would be 1.

      Therefore, either N is prime or N has prime factors greater than P.

      When I was 13, my maths teacher told the class that nobody had proved that the primes are infinite. I’m convinced he was really just challenging the brighter ones among us to find the proof. Sadly, none of us picked up on it.

  2. > nature can be expressed in mathematical rules because nature (or at least physics) is regular

    I like the argument. There are a few interesting edge cases, though, particularly with non-Euclidean geometries. Most mathematical rules reflect the real universe, so I agree that they were discovered. The branches of mathematics that extend known mathematics into non-traditional directions, I wonder on a case-by-base basis whether they reflect the universe itself, or they simply reflect our mathematics. It was fascinating to watch how imaginary numbers, which people may have thought had very little real-world application, have been useful when dealing with electrical circuits.

    Still, I might use separate terminology for mathematics that were derived using the scientific method and mathematics that were not, just as epistemologists might distinguish between scientifically justified knowledge and simple true facts that are a product of matauranga maori. Maybe instead of focusing on ‘invention’ or ‘discovery’, I would focus on ‘induction’ or ‘deduction’.

    1. “Most mathematical rules reflect the real universe, so I agree that they were discovered.”

      Right, but were those rules invented by humans because they allowed a useful modeling of the real universe? To me it seems like mathematics are wholly invented by humans. They are something like a language and system of logic invented by humans to describe the patterns of the real universe. Insofar as premises and rules of a given math correspond to reality, it seems to me that they do because humans made them specifically to do so. And that this evolves over time so that maths we’ve invented to model patterns we observe in the real world can reveal aspects of those real world patterns that we had not known of before. But maths can be invented to model completely made up things, or even whole universes too.

      1. Seeing that people invented numbers and mathematical rules, like a language, is strengthened by considering that there are some aspects of math that do *not* model the real world; geometries with different amounts of curvature (+, -, 0); imaginary numbers (for some aspects of reality, maybe any aspect, IDK how imaginary numbers are used in physics).

      2. “[ mathematical rules] are something like a language and system of logic ”

        (Took a while but this just occurred to me – this was the first comment with the word “language”, so I reply here )

        I think everyone regularly reads/hears about how mathematics is a language – of its own, of Nature, or otherwise.

        But many things can be called something like a language, with rules, and logic.

        Simplest one that comes to mind is music (I know I know! I can’t put it down!). There is written music, cadences, rules of harmony, etc.

        Other areas if science can, at some point, be remarked upon to have “the language” – that is, its practitioners know the language in the practice of that science – spoken or otherwise.

        So I am set to reject the common refrain that mathematics unique because it is like, or is, a language, because the same can be pointed out for unrelated areas of knowledge.

        I think Linguist suggested a difficulty rationalizing mathematics that way as well – I’m just noting my thought process, is all.

    2. Sounds a lot like my tentative view – that mathematical objects/properties exist, evidenced by the fact that our best scientific theories seem to require them. E.g., general relativity says that there’s a specific geometry to spacetime. If there’s a geometry to spacetime, then there’s a geometry. But it remains open to claim that mathematics that doesn’t get applications in science is pure invention.

    3. Most mathematical rules reflect the real universe

      I don’t agree. Most mathematical rules were obtained by mathematicians exploring the abstract world of maths. Some of them turn out to be useful for modelling the Universe or bits of it, but maths is its own thing not constrained to reflect the real world.

  3. I’m a determinist and a naturalist (i.e. believing that only the natural universe exists), so my first question is where would numbers exist? If you postulate some non-natural ‘space’ then anything *could* exist but you always have to explain how these non-natural spaces and contents interact with the natural world.

    1. The traditional response is to question pi. In what sense can it be said that pi is real? You can’t have a collection of pi objects the way you can have a collection of six objects. Does the universe require the pi to have its precise value? I’d argue yes, its value is absolutely constant, not only in Euclidean geometries, but probably also in non-Euclidean geometries (although I’m not familiar with the latter.) Would pi exist as a real number, even if the universe did not have any sapient life? I believe so.

      This is where it comes back to Platonic ideals (see the first line of the post). At that point, it is not necessary for numbers or other ideals to exist any non-natural space; they simply are. 1=1. The existence of numbers is tautological. They don’t need to be saved in the processor register of the universe.

      Mandatory XKCD: https://xkcd.com/435/

      1. In fact, if you define pi to be the ratio of a circle’s circumference to its diameter, it does not have its “usual” value (3.1415…) in non-=Euclidean geometries.

        For instance, on the surface of the (assumed-spherical) Earth, a circle made up from a line of latitude has a circumference:diameter rato of

        c:d = pi * sin(theta)/theta

        …where theta = (90 – latitude)*pi/180 and “pi” refers to the “usual” value. At the poles c:d = pi as in flat geometry. But at the equator, c:d = 2.

    2. As a naturalist myself too, I may recommend to you Santayana’s Scepticism and Animal Faith (1923). All qualia, all character things assume in our minds do not exist (only matter exist) but are. Our consciousness is pure actuality of matter, it exists but it is not material (being pure act, pure vision). This vision of mind as pure act is not the same as mind as epiphenomenon. Mind is not added to body (matter), but it is body (brain) in action, as pure act (mind). Where are numbers, words, qualia we see in our minds? In the logical realm of essence. This realm is part of nature. It is not beyond or behind or under nature. This logical (aesthetic) realm of essences, of qualia, is the form matter takes for us as it moves.

      The number 2 is not in our minds. The number “2” is the sign we see on paper. The meaning of that number must be understood in the our minds to interpret what is on the paper. Numbers are in the logical realm of essence our minds see, interpret, understand. They are, but they do not exist.

      When one confuses essences (universals) with existences (material particulars) come disillusion and myth and metaphysics (theology and idealism).

      1. “The number “2” is the sign we see on paper.” – In the philosophy of mathematics it is important to distinguish clearly between number-signs or number-words aka numerals (such as the numeral “2”) and numbers (such as the number 2), because according to mathematical platonism numbers are invisible abstract objects, as opposed to visible tokens of numerals. Note that there is another important distinction, namely the one between abstract types and concrete tokens of them. The numeral-type is an abstract object like the number 2; only the numeral-token “2” you see here is a concrete object located in space(time).

      2. I think the analogy between mathematics and language is apt. Language does not refer directly to the world but to human concepts (which are based in part on experience but are essentially a human model of the world). Similarly, mathematics is a human model of relations in the world.

        1. I was toying with the language/math analogy, too, but it breaks down because there are so many ill-formed linguistic constructs, meaningless terms that are frequently abused, everything from ‘god’ to ‘evil’ to ‘should’.

          Sure, math has divide-by-zero errors, but I would hesitate to say that math has any equivalent to the gobbledygook we hear from politicians on a daily basis. If we saw a string like “3..4 +* 9,2 = 5 = 3” most of us would not consider that a mathematical construct, the output of a mathematical system. Yet in real life, we encounter word salad and still consider it to be a use of language.

  4. Great question! Discovered it (math), I say. Or what I call “the Tom Brady Effect.” In an evolutionary framework it (math) is the ability to kill from a distance. Which propelled us to the dominate species on the planet.
    Those early, THAT person, who could CALCULATE speed, distance, velocity, etc. and so on enabling him (most likely) to kill from a distance … had a shit load of descendants. And had no idea of formal mathematics. Said another way: just an evolutionary “accident”.

  5. Reality can be described with the help of mathematics.
    Discovered or invented … ???? I am trying to see how my worldview changes should I come down on one side or the other.

    Somebody here had a great quote from Gödel:
    “The more I think about language, the more it amazes me that people ever understand each other.”

    1. “The more I think about language, the more it amazes me that people ever understand each other.”

      Yeah, I know what you mean.

  6. With all the recent ” Endtimes are near” because blah blah blah permeating too many airwaves commandeered by evangelical “leaders”, politicians and pundits lately, it would be interesting to hear their explanations for why and how Vedic writings pre-dating their sources’ claims by thousands of years present more rational mathematical explanations of how there could be at least a few more gods managing the universe. Of course, the best math problem (attributed to quite a few famous and not-so-famous) respondents is: “You’re almost as atheistic as me, except I just subtract one more god than you”.

  7. Numbers are signs for things, never the things signified. It is the same with numbers as with words. We invent them but they have a natural function: the success of this relation “sign-thing signified” via language gives the latter the appearance of magical power. The same happens with numbers. This success in the tool should not make us ignore that numbers are human tools to live better in nature, because nature can be mathematically understood, but nature is deeper than our theories about her. Our mathematical “discoveries” are our inventions, our conjectures, as Popper said.

    But not only numbers show that there is a mystery. Words show there is a mystery in the universe. Religions are based on this fact. Mysticism. Heidegger based all his philosophy on the meaning of being and Dasein. As knowing the meaning of nature is impossible, Heidegger ended in mysticism. “In the beginning was the Word,” said one Greek mystic-gnostic. Now Hermeneutics says that science is just limited and interpretation, rhetorics and communication (dialectics) truly human. Words, words, words over numbers, numbers, numbers…

  8. Our math models are approximations of reality. Okay, reality is regular, but we are only getting close to describing that reality. If we were to communicate with an alien civilization with a complete set of solutions to physics problems, I believe we would find strong agreement with counting, i.e., integers, but not necessarily with anything else. They may have completely different methods for representing reality. I once helped my daughter with a physics course in college. I noticed that it solved everything without calculus. I then realized there are no inherent methods for modelling reality.

    Of course, I feel that way, I’m an applied mathematician.

  9. Oh no – this is too much fun – My 2pc.

    Discovery is the identification of things that were there the whole time. So yes it was discovered.

    Beethoven’s 9th, Michaelangelo’s David, the English language, were not discovered. They are products of experience.

    DNA, the cosine function, black holes, exist independent of our experience. They had to be discovered – or we would not know about them.

    1. “Beethoven’s 9th, Michaelangelo’s David, the English language, were not discovered. They are products of experience.”

      I would say that Bach discovered more than created…

      1. Bach’s Three-Part Inventions did not exist before Bach. Bach invented them. People write in the Bach style – which he also invented. And even then, the music does not exist except when pressure waves impinge on an ear – an individual experience.

        But I jumped the gun – the game today is “discovery or invention”. But couldn’t resist Bach’s inventions. Because he named them inventions.

        1. “Bach invented them.”

          Disagree. I think that there’s something objectively ‘real’ about great art, and that even aliens could potentially appreciate Bach, or even create something analogous without knowing anything at all about Terran music.

          “…the music does not exist except when pressure waves impinge on an ear…”

          Disagree. It exists as an abstraction.

          1. Then you disagree with Bach.

            Bach wrote music to be heard – not cogitated upon. Bach certainly cogitated upon music unlike anyone else but that is the horse – the carriage is that we all hear it.

            1. “Bach wrote music to be heard…”

              To hear music is obviously more than “..pressure waves impinging on an ear…” And the intention behind writing music doesn’t settle the issue of what it means for music to ‘exist.’

              “Bach certainly cogitated upon music unlike anyone else but that is the horse – the carriage is that we all hear it.”

              But the horse still exists, with or without the carriage.

              “Then you disagree with Bach.”

              Did Bach express an opinion of Platonism?

              1. “Did Bach express an opinion of Platonism?”

                Bach wrote the compositions Two and Three Part Inventions and titled them for a reason.

              2. [ apologies for going on but this just gave me an epiphany of sorts ]

                “To hear music is obviously more than “..pressure waves impinging on an ear…” ”
                How can music be anything but that? I think that is _precisely_ what makes music so fascinating and mysterious – that so simple a phenomenon can give rise to such a richness of experience.

                [ we now return to our regularly scheduled programming ]

              3. [Not able to reply below.]

                “Bach wrote the compositions Two and Three Part Inventions and titled them for a reason.”

                And Bach also said: “I play the notes as they are written, but it is God who makes the music.” But it’s all kinda beside the point, since Bach is hardly the final word on this old debate.

                “How can music be anything but that?”

                So when an insect hears these vibrations, that is music?

        2. “Music does not exist except when pressure waves impinge on an ear”?
          I do not think so, music can spook in your head without pressure waves at all. It is a process in your brain that can be triggered by pressure waves impinging on your ear.
          Like seeing, you can see in a dream without light waves/particles impinging on your retina, it is a process in your brain.

  10. The question makes me wonder how and when maths was invented/discovered. Before that happened, the patterns of nature surely existed, but were not described in a precise way. Somebody at some time probably reflected on some regularity – perhaps the fact that she had the same amount/number of fingers on each hand. I have read – I think in Jared Diamond’s Guns, Germs, and Steel – that some primitive cultures, number is not precise but is thought of as: some, more, lots, humongous, etc. I suspect though the earliest numbers were used for counting such as the Sumerian agricultural records. More elaborate maths probably were relationships that were discovered to exist. Not that the numbers preexisted, but that the relationship between numbers was discovered to exist. Some complex math has no known utility or relationship to the real world that we know of. They are just thought invented to please the mind of mathematicians.

  11. [ wanted to add 1pc to my 2pc ]:

    Gears, transistors, scissors, are inventions and bear the property of getting old, worn out, and in need of repair or improved design. They are not discoveries.

  12. There are two types of mathematics: pure and applied. In many universities, this split is reflected in two separate departments, or at least, in a rather distinct dividing line in a single department. Most mathematicians identify as either pure or applied.

    The pure branch is concerned with the underlying structure and logic of concepts such as numbers and sets. It’s this branch which tries to answer questions such as “Can we prove that 1+1=2?”. The answer, by the way, is yes. Bertrand Russell and Alfred North Whitehead did so in their monumental Principia Mathematica a century ago. (Don’t ask me to explain it. I’m an applied mathematician.)

    The applied side uses maths as a tool to try to model the observable universe. The fact that this is feasible is, as PCC(E) notes, because the universe is regular. Mathematics is simply the most elegant and effective way to describe that regularity.

    As to whether humans invented mathematics or discovered it, I’m inclined to the latter view. I’m pretty sure that if we ever made contact with an intelligent alien species, they would recognise our mathematics as identical to their own, albeit that they would write it with different symbols, just as Newton and Leibniz devised different notations when they independently discovered the differential and integral calculus.

  13. …To the mystery of prime numbers, my own response would be, “Well, that’s just the way it is.”

    Ask a further question: “Could they be any other way?” Are the properties of primes forced by the way the universe is? So “that’s the way they are and they could be no other way” suggests they were discovered.

  14. For a materialist there is no mystery, what we call math are weakly emerging phenomena; in our case emerging from our brain processes. Mathematical realism is a kind of believe in miracles. It’s not in the slightest about what we observe, it’s only about what we believe.

    Fun fact. Humans are not the only animals that are able to count (f.i. chickens). There are even plants that count (Venus flytrap); so brains are not essential for doing math. Darwin discovered some time ago the process that made this possible.

  15. Hi! I’m a research mathematician and college professor.

    While I’ve never been interested in the “created or discovered” debate, I will say that my daily practice of doing research and proving new theorems makes it seem pretty obvious to me that math is *discovered*. When I prove a new theorem, it feels nothing like my own creation, and entirely like something which was waiting to be uncovered.

    To put this another way, I would say that the Pythagorean Theorem *is* somehow floating out there in the cosmos, in the sense that extraterrestrials who had developed a mathematical system with the same basic axioms as ours would also discover the Pythagorean Theorem (though they surely would name it after one of their own kind!).

    If you want to argue that we invented the axioms, well, okay, but just as life on earth evolved from simple primordial forms (and were not created in their present form), the mathematics that we have is a direct and unalterable consequence of our starting axioms. If you change the axioms, you change the mathematics, but after changing the axioms, you don’t get to choose any of the other changes — they’re decided for you, waiting to be discovered.

    Having said all that, this is not a hill I’m willing to die (or even stub a toe) on.

    1. Arguing from the other point of view, it seems to me that the Pythagorean theorem is a pattern in the real world and the math used to describe it is a made up collection of symbols designed by people to model the pattern with as much fidelity as possible. But we know that many of our best mathematical models do not correspond precisely with reality. Even the most well verified, to umpteen decimal places, mathematical model, the Standard Model + QM, is pretty clearly not entirely accurate. According to experts.

      Math seems similar to philosophy to me in that it can be an extremely powerful tool for understanding our reality, as long as it stays in touch with reality on occasion via empiricism. But if it doesn’t stay in touch with reality often enough, though the patterns you discover may be fascinating they are very unlikely to correspond to reality and aren’t useful in helping to understand it. Well, accept for ruling out ideas that don’t work, which is actually quite useful.

      And though I’m not a mathematician I did stay in a Holiday Inn last night! That is to say, it’s certainly possible that I’m completely wrong.

    2. Something of a counterpoint: In the standard “definition, theorem, proof” cycle, I’d have a difficult time believing that we are “discovering” a new definition — especially if the newly defined mathematical object is markedly different from the objects that came before it. Instead, it seems to me that we *create* a new definition and then *discover* what can be done with this new mathematical object.

      But I don’t have strong feelings on this.

      1. “Instead, it seems to me that we *create* a new definition and then *discover* what can be done with this new mathematical object.”

        I think you’re spot on. There seem to be two senses of discover here: one that is akin to digging in the ground and finding buried treasure, and a second one that consists in the implications of a conceptual framework that arose out of counting and its representation. Discovering buried treasure is not implied by the act of digging in the ground. It is contingent on the treasure being there. Statements in mathematics are necessarily implied by its conceptual framework (i.e., by definition) and, therefore, not discoverable in the sense of finding buried treasure.

    3. “Discovered vs invented?” How about “understand”? Being of the agnostic persuasion I rail against the word “know” but paraphrasing Sagan: We are a way for the universe to understand itself.

  16. I think it’s clear that the specific processes of mathematics and numbers are devised by humans, but the things they describe certainly seem to be inherent, discovered characteristics of the universe…and quite possibly all universes.

    In that sense, the Pythagorean Theorem WAS just floating out there…not literally, but it was always implicit in the geometry of Euclidian planes, wherever they might occur. Ditto with the primes…they are there, implicit, in the characteristics of reality, and it’s difficult to see how any reality could be otherwise. As someone once said, “We can imagine universes in which any and all of the constants of nature might be different, but we cannot imagine a universe in which there are a finite number of prime numbers.”

    Any universe that is self-consistent would, as far as I can see, have to have, implicitly the laws of basic arithmetic (by whatever names or symbols anyone might signify them, or even if there is no one to do so), and Euclidean and non-Euclidean geometry, whether instantiated in anything in particular or not, and all the things logically implied by these, even if never instantiated in any natural process, though in that case they would very much be only POTENTIAL.

    In that sense, at least, I guess that means I think that math is real and predates people, but I don’t imagine some realm somewhere that contains Platonic ideals or anything of that sort. I just think that any “possible” universe has to be internally non-contradictory, and given that, for instance, what we call 2, what we call +, what we call 2 must what we call = what we call 4, (no matter what the Party or Big Brother says). And the ratio of the circumference of what we call a circle to its diameter is always going to be that number we call pi.

    1. [W]e cannot imagine a universe in which there are a finite number of prime numbers.

      Oh bother — I managed to get almost all the way through the thread without commenting, but my Inner Mathsie just wouldn’t let this go.

      The fact is that there are otherwise-normal mathematicians who do imagine such things, using common mathematical practice but with slightly different axioms and inference rules; these folks are called “strict finitists”.

      It just so happens that one can have consistent systems of numbers without allowing arbitrarily large ones. In such a system there is no “successor” axiom (which in usual arithmetic requires that every number X has a successor X+1). Weird, eh?
      See https://en.wikipedia.org/wiki/Ultrafinitism for an introduction.

      As a rough generalisation, we cannot imagine a correct bound to the range of things that mathematicians might imagine.

      1. Well, to be fair, though you are obviously correct, I meant to imply “what we mean when we talk about prime numbers in general” when I said we can’t really imagine a universe in which there are a finite number of prime numbers. And I really like the last sentence of your comment particularly! ^_^

        1. Thanks.

          And of course I fully agree that in normal arithmetic the rules of mathematical proof force the conclusion that there is no largest prime number. But we humans do imagine all sorts of inconsistent universes, sometimes three times before breakfast, since we are not constrained to follow strict rules of mathematical proof. Imagining fictions comes naturally (“The devil made me do it!”); imagining consistent ones definitely does not (which is why good science fiction writers make the big bucks 🙂 ).

  17. I guess my question is, “could we have invented a different mathematics?” I don’t think so. Sure, people could use base 12 or the like, but that is just different ways to get to the same place.
    I think it is an entirely descriptive way to understand that which has always existed.

  18. Discovered, not invented. In fact, I’d say that the burden of proof is on those who’d claim that anything other than math actually exists…

  19. I forget who it was – Wigner perhaps? – who talked about the “unreasonable effectiveness of mathematics”. But I remember a YouTube interview with the mathematician Stephen Wolfram where he points out that there are some things which maths is not so good at modeling; what we understand is mainly the stuff that it is good for.

    1. From mathworld.wolfram.com/PrimeNumber.html

      A prime number . . . a positive integer p>1 that has no positive integer divisors other than 1 and p itself . . . The number 1 is a special case which is considered neither prime nor composite . . .Although the number 1 used to be considered a prime . . . it requires special treatment in so many definitions and applications involving primes greater than or equal to 2 that it is usually placed into a class of its own.

      A good reason not to call 1 a prime number is that if 1 were prime, then the statement of the fundamental theorem of arithmetic would have to be modified since “in exactly one way” would be false because any n=n·1. In other words, unique factorization into a product of primes would fail if the primes included 1. A slightly less illuminating but mathematically correct reason is noted by Tietze (1965, p. 2), who states “Why is the number 1 made an exception? This is a problem that schoolboys often argue about, but since it is a question of definition, it is not arguable.”

      That last sentence sounds like a “just so” statement to me. I’ve thought that I’ve dealt satisfactorily with addition since I was a “schoolboy,” but now it seems I need to delve into the fundamental theorem of addition to understand why 1 is not a prime number.

      As I recall, I was taught that a prime number was any number the only possible factors of which were 1 and itself. To my mind, in the case of 1 the other number happens to be 1. (What other integer can be multiplied with 1 to equal 1?)

      If a 5th grade student asks a teacher to specifically demonstrate why 1 is not a prime, is it satisfactory for the teacher to reply that “since it is a question of definition, it is not arguable” and that the student should consult the fundamental theorem of addition for further clarification?

    1. Who does the counting?

      My belief is that counting, numbers and defining what is counted is a human cognitive thing (Yes, some other animals also seem to count). And, as alluded to by the earlier George E.P. Box, no model is a full and true representation of reality.

  20. Oh dear! This stuff is endlessly fascinating. I think that the mathematics of relationships is already there, but not articulated until we choose to articulate it in symbols.

    Nature somehow knows of the distance between two blades of grass or two trees or two continents.

    Prime numbers are the subject of Gods. They are astonishingly difficult to work with. I imagine that there is order inherent in them but that we do not yet have the tools to understand them properly and to us they appear to have random as well as deterministic properties.

    One fellow who was pretty good with numbers was Albert Einstein and he said of Mozart’s music that, while Beethoven created his music, Mozart’s music was so pure that it seemed to have been ever-present in the universe, waiting to be discovered by the master.

    Is music already there, waiting to be discovered? In a sense yes and, then again, no. I trust I make myself clear!!!

    In galaxies far away lurk civilizations that are as fascinated with primes as we are. They, too, sense the notions of distance, structure, similarity and dissimilarity, number, discrete, continuous, rates of change etc and they too take operations such as addition and division for granted, as we do. Their mathematics will flow from such premises, as ours has done. But they will need something akin to writing in order to articulate their mathematics. But, in aggregate, their mathematics will be similar to ours. Nothing much changes under the sun – even under the suns of far away worlds!

    Similarly, they will sense Newton’s Laws, electromagnetism, thermodynamics, optics, solid state physics, nuclear physics and general relativity, much as we do. They might do something akin to meteorology and dynamic oceanography (one of my own favorite subjects of my Masters Degree many years ago – but then all of the above branches of Physics were fascinating to me as a young man). And so forth!

    The Goldbach Conjecture has not yet been proved. Every even number greater than 2 is the sum of two primes (e.g. 18 = 11 + 7 and 22 = 17 + 5 ). Similarly, the Twin Prime Conjecture (the number of pairs of adjacent primes separated by two is infinite) has not been proved and it may take decades to do so, if not much more.

    I wrote a layperson’s article on primes a few years ago. Here it is:

    https://www.royalsocietyofnewzealandwellingtonbranch.org/blogs/a-quick-look-at-prime-numbers

    In there I said that the first 15 gaps between adjacent primes are as follows:
    1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4 and 6.

    I said that we see that the even natural numbers 10, 12, 16 and others within the range of this list are not known gap sizes. This was incorrect – a residue from not deleting something that I had written earlier. I should have said something there about Goldbach but I did not.

    I am not religious in any way but it is through prime numbers and through the music of Wolfgang Amadeus Mozart where I have come closest to sensing God and touching the infinite.

    David Lillis

  21. I once had an argument with a guy who insisted that “2+2=4 is meaningless in the real world; it’s just something that we say because it’s convenient ” I think the fact that different cultures have come up with different ways of multiplying and dividing and get the same results shows that numbers exist independent of human invention (12 x 3 equals 36 regardless of whether you are English, Chinese or Aztec). Of course,
    some math is invented, such as measurements (12 inches to a foot): so it’s both.

    1. You can’t base arithmetic on logic axioms, so you have to derive them – including multiplication – because you find them useful. Every culture that developed farming would find use for methods to compare areas.

  22. Math is obviously an invented toolset. A test for that is observing the many math areas that are just curiosities and have found no empirical use, there is no 1-to-1 map between these and nature. Moreover we know that physics such as quantum field theory cannot be made into axiomatized form.

    Physics laws are based on symmetry and symmetry breaking as shown in Noether’s theorem relating them to conservation laws. We wouldn’t be here if they didn’t exist, so asking why they exist is akin to asking why we exist.

    1. [ can’t … resist … ]

      “Math is obviously an invented toolset.”

      I don’t think that excludes it as being comprised of discoveries.

      “A test for that is observing the many math areas that are just curiosities and have found no empirical use, there is no 1-to-1 map between these and nature. ”

      May I introduce you to the useless machine

      https://en.m.wikipedia.org/wiki/Useless_machine

      … but after churning this over in my head during chores just now – fittingly using inventions to do them – pairs of scissors, the English language, the number zero, nuclear magnetic resonance, storytelling, ad infinitum, are conglomerations of both distinct inventions – new concepts or objects which did not exist until our production of them – but also overlap with clear discoveries – explaining a phenomenon or problem in terms of something that already existed prior to having been found.

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