Do Mathematical Objects Exist?

I am slowly working on an article for Skeptical Inquirer about the ways in which religious apologists use mathematical arguments in their rhetoric. Among these arguments are the familiar creationist claims about probability and information theory, but there is also a family of arguments based on the effectiveness of mathematics itself. The basic argument is that mathematics is so useful for describing the world solely because God, in his benevolence, designed the world to be describable in that way. They will often cite Eugene Wigner’s 1960 article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” as supportive of their basic view.

In pondering what to say in reply to this dubious argument, I’ve been reading some philosophy of mathematics lately. Inevitably, the question of whether mathematical objects exist has arisen. So, in the spirit of my ongoing discussion about things that frustrate me about philosophy, let’s consider that question now.

Here’s mathematician Timothy Gowers, a Field’s Medalist, from his book Mathematics: A Very Short Introduction:

A few years ago, a review in the Time Literary Supplement opened with the following paragraph:

Given that 0 x 0 = 0 and 1 x 1 = 1, it follows that there are numbers that are their own squares. But then it follows in turn that there are numbers. In a single step of artless simplicity, we seem to have advanced from a piece of elementary arithmetic to a startling and highly controversial philosophical conclusion: that numbers exist. You would have thought that it should have been more difficult.

This argument can be criticized in many ways, and it is unlikely that anybody takes it seriously, including the reviewer. However, there certainly are philosophers who take seriously the question of whether numbers exist, and this distinguishes them from mathematicians, who either find it obvious that numbers exist or do not understand what is being asked. The main purpose of this chapter is to explain why it is that mathematicians can, and even should, happily ignore this seemingly fundamental question.

The absurdity of the `artlessly simple’ argument for the existence of numbers becomes very clear if one looks at a parallel argument about the game of chess. Given that the black king, in chess, is sometimes allowed to move diagonally by one square, it follows that there are chess pieces that are sometimes allowed to move diagonally by one square. But then it follows in turn that there are chess pieces. Of course, I do not mean by this the mundane statement that people sometimes build chess sets — after all, it is possible to play the game without them — but the far more `startling’ philosophical conclusion that chess pieces exist independently of their physical manifestations.

What is the black king in chess? That is a strange question, and the most satisfactory way to deal with it seems to be to sidestep it slightly. What more can one do than point to a chessboard and explain the rules of the game, perhaps paying particular attention to the black king as one does so? What matters about the black king is not its existence, or its intrinsic nature, but the role that it plays in the game.

This seems exactly right to me. Mathematical objects have precisely the same kind of existence as the pieces in a game of chess. The king has a role to play in the game of chess, and it has certain attributes bequeathed to it by the rules of the game. In that sense, it exists. But if you then say, “No, no. I want to know if the black king exists independently of anyone’s ideas about how to play chess,” then I no longer understand what you are talking about.

Another way of making the point is that questions about mathematical objects are essentially identical to questions about fictional characters. If someone asks, “Did Sherlock Holmes smoke a pipe?” then in one sense the answer is just no. Sherlock Holmes never existed in the real world, so obviously he didn’t smoke a pipe. But if we understand that question to be referring specifically to the fictional world created by Arthur Conan Doyle, then the question is meaningful. Moreover, within that context it makes sense to answer that yes, he did indeed smoke a pipe.

Likewise for mathematical questions. Mathematical objects exist in a fictional world created by mathematicians. For example, in the context of modern abstract algebra I know what it means to ask if finite, simple groups exist. But if you ask whether finite, simple groups existed prior to the establishment of modern abstract algebra, than I don’t know what you mean. It’s like asking if Sherlock Holmes existed prior to the writings of Arthur Conan Doyle.

There are obvious parallels here with our earlier discussion of realism vs. anti-realism in the philosophy of science. Perhaps there are some analogies we can find to provide guidance here. A scientific realist would say that electrons, say, existed before scientists ever thought to formulate the concept. Maybe we can say the same about prime numbers; they existed prior to the time when mathematicians thought to formulate the concept.

But this doesn’t help. The premise of scientific realism is that there is a physical world that exists independently of anyone’s ideas about it. While I would not want to have to write down a rigorous definition, it seems clear what we mean when we talk about the existence of physical objects. Not so with mathematical objects. Whatever sort of existence they have, they certainly are not part of the physical world. Mathematical objects just are whatever mathematicians say they are.

What about the usefulness of mathematics? The main reason for thinking that electrons really exist out there in the world is their incredible usefulness in the context of scientific explanations. It would be a miracle if they were so useful without actually existing. Likewise for mathematical objects, we could argue. What a miracle it would be if math were so useful, even indispensable, in the context of scientific explanations, if they did not actually exist.

But this argument also does not seem very persuasive. There is no more mystery in the usefulness of mathematics than there is in the usefulness of maps in navigating unfamiliar areas. The map is not the territory, but the pattern of dots and squiggly lines on the paper is inspired by the territory. Likewise for the particular abstractions mathematicians choose to study. A perfect circle has no existence outside of a geometry textbook, but our abstract notion of a perfect circle captures something important about certain physical objects. The abstract models of mathematics are like the abstract version of the territory presented on a map. They help you see reality more clearly without themselves being part of that reality.

The general view of mathematics I am defending is known as fictionalism. It’s main rival is Platonism, which holds that mathematical objects exist independently of anyone’s ideas about them. It might be objected that some of what I have said in this post is question begging. For example, I said mathematical objects just are whatever mathematicians say they are. But that’s precisely the point at issue.

It’s not that I am specifically trying to beg the question. It’s just that I don’t know how else to talk about mathematics except in the context of fictionalism. My reply to someone who insists that mathematical objects exist in some non-spatio-temporal realm that we come to understand through mathematical research is just to stare at him blankly. I don’t understand what he’s trying to convince me of. The existence of physical objects makes sense to me. The existence of abstract objects that cannot possibly be reduced to some physical phenomenon does not make sense to me. The concept of existence does not seem helpful here.

Hence my frustration. Mathematical Platonism has some passionate defenders, but the position makes no sense to me. More importantly, though, my objection to this argument is the same as my objection to the argument between realists and anti-realists in science. I don’t see how anything at all is riding on this, and I don’t believe that anyone is thinking more clearly about mathematics as a result of the point and counterpoint in this discussion.

Of course, this is far more high-falutin than anything the religious demagogues have in mind. So, in the interest of ending on a lighter note, here’s a quotation from Katherine Loop, from her book Revealing Arithmetic: Math Concepts From A Biblical Worldview:

For example, if we use counting to explore our hands, we find we have four fingers and one thumb on each hand. Counting helps us see the design God placed within hands! At the same time, though, if we were to count the fingers on every person’s hands, every once in a while we would find a person with fewer/more fingers than usual, evidence we live in a fallen universe.

See my earlier remarks re staring blankly.

Comments

  1. #1 Valhar2000
    October 2, 2012

    See my earlier remarks re staring blankly.

    I know how you feel.

  2. #2 Zachary Alain
    http://zachmakesupstuff.blogspot.com/
    October 2, 2012

    My view is something close to `formalism’, i.e. that the existence of mathematics is like the existence of programming rules for Matlab. I don’t have a very big problem with mathematical Platonism, since I don’t think it of any great consequence. “There are correct rules of syntax for programming languages which do not exist.”

    As to the question of why mathematics `works’, I think there is a double answer: (1) we select axioms and structures that are useful, and (2) that principles in a universe in which we can exist can be captured mathematically is unsurprising.

    (2) is the only part of the answer on which it is worthwhile to elaborate. One can approach this from the usual direction of probability theory, but that would be uninteresting. More interestingly, Ramsey theory can be plausibly interpreted as telling us that we are guaranteed to find complicated structures in universes with lots of stuff. (And if we view our universe as a sort of particularly interesting subgraph…) Another way is to focus on something like universes with general laws of motion, particularly laws of motion which constantly act on e.g. particles. We can interpret the `constant action’ there as saying that the law of motion, say expressed as a function f(t), is iterated infinitely many times in any non-empty time interval. Or let it the function iterate at discrete intervals. Will any such function do? I’ll leave this approach – if it works – to people who have studied dynamical systems.

    To me, the `capture’ of real-world stuff by strictly abstract objects is unsurprising. Many of the structures, like groups, fields, and vector spaces, which crop up in physics do so with little to no coincidence. Groups describe symmetries. So the question is, why do symmetries exist? We can babble, or do hard stuff.

    Maybe I would be more impressed by the `coincidentalness’ of mathematical power if I had a solid background in mathematical physics. I’m just having trouble conceiving of any plausible universe in which (1) humans can exist and (2) mathematics cannot be useful.

    On a lighter note, read this very inadequate Wikipedia entry.

  3. #3 Cubist
    San Francisco Bay Area, California
    October 2, 2012

    The proper answer to “why is mathematics so unreasonably effective?” is, “why *shouldn’t* mathematics be so unreasonably effective?” Math can be thought of as an über-generalized descriptive language. If you’re working in a universe that’s flat (Euclidian), math can describe it just fine; if you’re working in a universe that’s spherical (non-Euclidian), math can describe it just fine; if you’re working in a universe that’s hyperbolic (a separate flavor of non-Euclidian), math can describe it just fine.
    Since math can describe pretty much *any* *conceivable* *universe* *whatsoever*, it makes perfect sense that math can describe the specific universe in which you and I happen to live. It couldn’t be otherwise, *unless* you and I live in the sort of universe which math *cannot* describe.

  4. #4 Tridecanol
    Switzerland
    October 2, 2012

    Well, numbers are… numbers.

    It doesnt mather how you call them or how you write them.

    If a few thousand years ago the arabs chose other symbols to represent them, it wouldnt change their nature.
    If you’d define, lets say, the letter U to represent the number 5 for the rest of mathematical history, that wouldnt change a thing about mathematics. Its still 9-4=5 (or in this case “U”).

    What we colloquially know as Numbers are, just like you said, just symbols for an abstract idea.

    Likewise, words (collections of symbols) dont really exist. They just represent the abstract and quite complicated biochemistry in our brains that make us “recognise” a real physical object.
    EG. If i write “You can sit down in that chair”, you cant really do that, because in this case the chair is just some symbols on a webpage or piece of paper. However you instantly recognize the physical object even without me pointing to a real chair.

    To finish my ramblings, I’d like to quote my former phylosophy professor.
    “If all of the sudden all humans on earth chose to call the color of the sky green instead of blue, the sky would still remain blue as the word used to describe certain features of an object is just definition.”

  5. #5 Bee
    Stockholm
    October 2, 2012

    I recently wrote about this topic too, this might be interesting for you

    http://backreaction.blogspot.com/2012/09/what-are-you-really.html

  6. #6 John Jenkinson
    http://www.johnjenkinson.me
    October 2, 2012

    I think some numbers are real that define our reality as we know it. More numbers and possibilities go beyond this and most western religions fail to go beyond our sensory attributes and therefore other possibilities. including numbers!

  7. #7 david
    October 2, 2012

    What does “exist” mean? Does it mean to have substance, and be made of matter or energy? That is the usual sense of the word: I can rap my knuckles on the desk, it is hard, it exists. In that sense, numbers don’t exist. Does it mean that somebody can describe it, in abstract terms? Then mathematical objects exist, but so do fairies and unicorns. I cannot think of a definition of existence that encompasses number that does not also allow such madness.

  8. #8 Neil Rickert
    October 2, 2012

    Several years ago I purchased the Benacerraf and Putnam book “Philosophy of mathematics: Selected readings” – partly because it was on a closeout sale and appeared to have some interesting articles.

    As I read a number of articles, I kept seeing a puzzlement over the question of ontology of mathematical objects. So I asked a philosopher (on usenet) why. And he carefully explained to me why ontology is important. Once I fully understood that, understood the importance of ontology to philosophy, I began to realize that philosophy itself made no sense.

    Like you, I’m a fictionalist.

  9. #9 Miles Rind
    Cambridge, Mass.
    October 2, 2012

    “At the same time, though, if we were to count the fingers on every person’s hands, every once in a while we would find a person with fewer/more fingers than usual. . . .”

    Especially in the Bible Belt.

  10. #10 couchloc
    October 2, 2012

    “Bernays observed that when a mathematician is at work she “naively” treats the objects she is dealing with in a platonistic way. Every working mathematician, he says, is a platonist….. But when the mathematician is caught off duty by a philosopher who quizzes her about her ontological commitments, she is apt to shuffle her feet and withdraw to a vaguely non-platonistic position.”

    This quote seems rather fitting in this context. A good account of the whole issue of mathematical objects can be found here for those interested. Remember that Godel and Frege were platonists and this view cannot simply be laid at the feet of philosophers.

    http://plato.stanford.edu/entries/philosophy-mathematics/

  11. #11 eric
    October 2, 2012

    I think cubist has it right. The effectiveness of mathematics is not particularly surprising (to me). If there is a physical world, things in it will have relationships. If you have a general language for describing relationships between things, its going to apply. I suppose that in a completely chaotic universe mathematics would not be expected to be useful. But in any universe with regularities, its going to be useful for describing those regularities because it can (in principle) describe any regularity.

    Having said that, I think its also worth pointing out that different mathematical formalisms are not equally effective at describing the world; the ones we use are a result of trial, comparison, and selection. For example, nobody does calculus in roman numerals. You might get the same result, its just harder; a less useful fit.

    So, the fact that modern mathematics is so good at describing modern theories of how the world works is no more surprising than discovering that a shovel is really good at digging holes. Of course it is; we built it that way.

  12. #12 eric
    October 2, 2012

    Couchloc:

    Bernays observed that when a mathematician is at work she “naively” treats the objects she is dealing with in a platonistic way

    This is not surprising, but it doesn’t really support the platonist position at all. When a baseball ump is at work, she naively treats the strike zone as a real thing too.

    In fact whenever we take some human cultural activitiy seriously, ‘taking it seriously’ almost always involves treating some artificial convention as if it were a real thing. The same could be true with lawyers treating laws as real or cooks treating health regulations as real.

  13. #13 Beth
    October 2, 2012

    One way we determine what is real and what is illusion is through concensus with other humans. If everyone sees a tree, agrees on the specifics of it such as location, size, etc. , then we conclude the tree is real and not an illusion or mistaken impression.

    It seems to me that numbers are real in the sense that others can sense them and agree on their specifics such as whether it is prime or composite, etc.

    I’m honestly not sure how to classify the existance of non-physical abstractions that are capable of being sensed by different humans who agree on their charactoristics.

    It seems to me that agreement amongst independent observers is an important charactoristic of ‘real’ things. But while this agreement is a necessary component of determining what is real, I’m not sure that it is sufficient.

  14. #14 Gingerbaker
    October 2, 2012

    “*…unless* you and I live in the sort of universe which math *cannot* describe…”

    I think that we do, in fact, live in a universe that math can *not* describe. Not perfectly, anyway, which seems to be the premise behind the apologist’s argument that math is “unreasonably effective”.

    The physical world is not arranged in a form that is consistent with linear mathematics. Linear mathematics will get you close – usefully close in a practical sense, but not spot on. And the same for non linear math – close but no cigar.

    The universe, in fact, is about as far away from being mathematically “intelligently-designed” as it could be. The mathematics needed to describe even simple biological functions is very complex. The real world is decidedly not mathematically elegant.

    Now, if the Catholic God had indeed intelligently designed the universe for mathematical elegance, everything in it would have trinitarian symmetry. We would have no need for any fractions besides 1/3, and the only exponent we would need would be the cube. No exponents needed if Yahweh was the designer.

    If the universe really was non stochastic it would be an argument for a divinity. But the apologists are, it seems to me, staking a false claim here – mathematics is not unreasonably effective, nor is the universe mathematically designed.

  15. #15 MattB
    October 2, 2012

    Unfortunately, you can’t handwave this away. If you’re going to claim the statement ‘black kings move diagonally in chess’ is only true where there are chessboards and meaningless otherwise, you’ve given ‘truth’ its own rather strange Platonic character – it has a changeable ontological status, coming into being when the rules of chess are codified and vanishing again after…what? The destruction of all chess sets? The death of the last conscious entity that knows how chess is played? The death of the last conscious entity that could conceivably reinvent chess? The only way out is to accept that ‘black kings move diagonally in chess’ will always have been true.
    For maths it gets stranger because the problem extends both ways in time. Not only does it appear that the universe will continue to behave in a way that is not just well-described, but described to a quasi-arbitrary degree of accuracy, by human-generated cognitive structures derived, albeit at several removes, from simple concepts of counting, but as far as we can tell it always has. There is no reason whatsoever that this should be the case. Not even the anthropic principle helps here – ocelots are perfectly capable of surviving in a universe that doesn’t appear to have equipped them to understand relativity.
    I remain a staunch atheist and logic dictates that substance dualism doesn’t work – so I find it hard to imagine that numbers ‘exist’ in some mysterious supernatural realm and the idea that we require some kind of magic entity to guarantee reality. I do, however, think that philosophical questions about the way maths appears to work and the accuracy with which it describes the physical universe can’t be dismissed as silly (they can, of course, be ignored as irrelevant to the matter at hand and doubtless most mathematicians do this). It’s a ‘hard problem’ in the same sense as consciousness – it may even be an element of the same ‘hard problem’.

  16. #16 proximity1
    October 2, 2012

    Try and put yourself in the place of those who are posing the questions– what are they actually asking? Consider that they may not know the actual correct answer to that question.

    Perhaps they are asking something like:

    What would exist if there were no beings capable of thought?, that is, if thought itself were not possible—what things could not or could “no longer” “exist”?

    For a very long time people have pondered the reality or unreality of “universals” or of Plato’s “forms”–a dualism in which a metaphysical form is the supposed reality behind all “representations” of such forms. Why did people come to think in that way and what, and why, over time, was this view abandoned—when it was abandoned, that is–in scientific practice? Maybe a review of that would help in an approach to others–and why, after all, wouldn’t there be many of them today just as there have been over the centuries?–who haven’t made that intellectual journey yet.

    If looked at in that way, you might arrive at a more interesting consideration of the issues than what you’re offering readers above.

    In addition, B. Russell wrote a good deal about such issues. One interesting essay is his “Non-demonstrative inference” in his book, “My Philosophical Development”; there is also the last chapter of his book, “An Inquirly Into Meaning & Truth” which bears on the issues. Both of those are well worth your giving time to read. There is something of reinventing the wheel in much of the essay above. Others have done this work and done it rather well. Maybe they should be consulted before you finish drafting that article for Skeptical Inquirer.

  17. #17 proximity1
    October 2, 2012

    P.S.

    It would also help, I think, if you considered the possibility that in more than a few realms of current science practice, the currently held paradigms, when examined very carefully, show they presume a dualist formalism, a version of Plato’s forms, or of Aristotle’s naturalistic view of the physical world. That, properly, should be recognized and rooted out, but instead, it’s generally not recognized and understood at all. So it passes into and through and through the theoretical assumptions used in the various flawed paradigms –and very often with quite serious harm as a consequence.

    More than a little of standard practical science today has a few hard lessons to learn in valid reasoning and in something else philosophy takes an interest in: humility and its proper place. Too many scientistst and mathematicians stop at mouthing peities about humility without ever taking them seriously enough to get acquainted with it in actual practice.

    Where is scientists’ humilty today? In my opinion, it’s very often AWOL.

  18. #18 Kevin r
    Texas
    October 2, 2012

    This seems to be a parallel to the question of do souls exist. Does a person have an identity that exists outside physical reality?

  19. #19 MNb
    October 2, 2012

    “Mathematical objects just are whatever mathematicians say they are.”
    Looks sensible to me. After all I can collect some objects of the same kind, count them and say there are 00110 of them or 6 and mean exactly the same.

    “What about the usefulness of mathematics?”
    When schooled to become a teacher maths and physics I defended the answer none. My teachers weren’t happy with this, but indeed until today I reply to my pupils “only as a tool for other fields of research”.
    So I tend to fictionalism as well.

    “Counting helps us see the design God placed within hands!”
    Wishful thinking.

    @Eric: “I suppose that in a completely chaotic universe mathematics would not be expected to be useful.”
    Maths doesn’t look useful to me in universes as described by Tolkien and Rowlings.

    @Beth: “numbers are real in the sense that others can sense them and agree”
    I can play a game of blindfold chess with JR. Then we can sense the 12 different chess pieces and agree on their specifics. Still the entire game still takes place only in our heads. So what we two imagine is real? Sounds like a non-sequitur to me.

    @MattB: “The only way out is to accept that ‘black kings move diagonally in chess’ will always have been true.”
    Which doesn’t mean anything either. The laws of chess were different more than 500 years ago.

  20. #20 proximity1
    October 2, 2012

    other reading references:

    “Gödel and the Question of the Objective Existence of Mathematical Objects”, in Cassou-Noguès “History and Philosophy of Logic”

    http://www.informaworld.com/openurl?genre=article&doi=10.1080/01445340500112124&magic=crossref||D404A21C5BB053405B1A640AFFD44AE3
    and

    “After Gödel: Platonism and Rationalism in Mathematics and Logic” in Cassou-Noguès “History and Philosophy of Logic”

    http://www.tandfonline.com/doi/abs/10.1080/01445340.2011.648336

  21. #21 Blaine
    October 2, 2012

    I would like to know why such a vanishly small amount of mathematics is applicable to the ‘real’ world if mathematical objects exist and are related to the world.

    We now suspect that at the very bottom of things, reality is asymmetrical and so we will never have a TOE that is a unified underlying mathematical theory that explains everything. Instead, it will remain a pathwork quilt of subfields loosely connected.

  22. #22 Daniel Lafave
    October 2, 2012

    You seem to be advocating a form of structuralism.

    http://plato.stanford.edu/entries/philosophy-mathematics/#StrNom

    Some very smart people already had this idea and have already worked out many of the implication of such a theory, so why is that something that annoys you about philosophy rather than something from which you can fruitfully learn.

    Philosophers care about ontology because they care about the truth of claims. If a mathematical claim is true, then it must be true in virtue of a truth-maker. What is that thing? Maybe that something are just structural facts about ordinary physical objects, but it does no good to just say, I’m going to ignore questions of ontology and pretend that the question isn’t there.

  23. #23 Jonathan Lubin
    http://www.math.brown.edu/~lubinj/
    October 2, 2012

    Even though you and I have had a little disagreement on related issues in the past, today’s piece seems to me perfectly on the mark.

  24. #24 Blaine
    October 2, 2012

    @Kevin r
    “This seems to be a parallel to the question of do souls exist. Does a person have an identity that exists outside physical reality”

    I am not so sure about this. Life and consciousness are processes not substantial things. There is no such thing as a ‘fixed’ identity. How many times have ‘you’ changed since you were a child? We are negentropic energy sinks. Ie., we consume energy and export entropy to the environment…in short, we are machines.
    This is an ancient discussion which goes back to Plutarch’s Ship of Theseus. Imagine repairing Theseus’ ship. Over time, every piece of wood is replaced as it is being repaired. When the repair is complete, is it the same ship? Now, take all the old wood and rebuild the ship. Which ship is Theseus’ ship? Is it the repaired ship or the ship constructed from the old wood?
    The gist of the issue in regard to mathematics is whether mental objects have external referrents. I can imagine my dog spot, and spot actually exists. After he dies he no longer exists. I can imagine flying unicorns but they don’t exist and have no external referrent. The issue with mathematic is similar.
    More recently, these issues have been explored at great length by semioticians. See for example:
    http://en.wikipedia.org/wiki/John_Deely
    If mathematicians would take the time to read the historical discussions of these issues, the level of the discussion would be higher. Its not that we wouldn’t be confused, but we would be confused at a higher level and confused about more important things.

  25. #25 Jason Rosenhouse
    October 2, 2012

    Daniel –

    My frustration is that when people ask if mathematical objects exist, I don’t understand what the question means. Existence is a concept that seems clear and useful when applied to physical objects, but it is not clear at all when you try apply it to purely abstract concepts. (I can accept a statement like, “Love exists,” but only because love is ultimately, if a bit cold-heartedly, reducible to physical concepts.)

    I m frustrated because philosophers do seem to think this is a clear and sensible question. For example, the Stanford Encyclopedia article on Platonism glosses over the question completely, describing it as “tolerably clear” what it means to say that mathematical objects exist. Well, it does not seem tolerably clear to me.

    Finally, I would say that it does a lot of good to ignore pointless pseudo-questions that do little to help us understand anything important about math. That’s why I like Gowers’s statement. What matters about mathematical objects is not whether they exist. What matters is what they do for us.

  26. #26 Beth
    October 2, 2012

    @MNb

    Does chess exist without the pieces? Is a mental chess game less real than a chess game played with physical pieces moved about on a board?

  27. #27 eric
    October 2, 2012

    Mnb:

    Maths doesn’t look useful to me in universes as described by Tolkien and Rowlings.

    Seriously? Both Hogwarts and Minas Tirith appear to be described as fairly structurally sound. Foundations imply someone did a caculation of load and settling. Otherwise, why bother with them? I vaguely recall the riders of Rohan talking about how many days’ ride one thing was away from the other. Figuring out how many troops they might need. How much food and grain and such. Wonder how they did all that when math was useless.

    And so on. The existence of miracles will perhaps make the fit between a mathematical model of reality and reality not as good. But as Gingerbaker points out, an imperfect or inaccurate model can still be useful.

  28. #28 MattB
    October 2, 2012

    @MNb
    Claiming a lack of ‘meaning’ just leads us into word games. It is either an eternally true property of the universe that at some point black kings have moved diagonally in chess – or truth itself is contingent on a statement being understood by a conscious entity and/or applicable to a really existing object, and we collapse into solipsism.

    @JasonRosenhouse

    “Finally, I would say that it does a lot of good to ignore pointless pseudo-questions that do little to help us understand anything important about math”

    It depends what you mean by ‘us’. If ‘us’ is mathematicians, then fair enough. If you mean it more generally then you imply the position that ‘understanding important things about math’ is the only worthwhile human activity. Which is something of a value judgement, to say the least.

  29. #29 Darth Dog
    October 2, 2012

    @MattB

    I don’t see how it can be an eteranlly true property of the universe that black kings move diagonally in chess. It only has meaning if there is such a game as chess that is conceived by someone. Otherwise consider all the games that have never been invented, and never will. There are infinitely many of them. And they have lots of rules. Are all those rules true also? Is it true that in the game of filbitz, which no one ever has or will conceive, that a fibrillator can’t motivate? That type of existence doesn’t seem like a very useful concept.

  30. #30 eric
    October 2, 2012

    Reading MattB and Darth Dog’s posts, it strikes me that the platonist approach is somewhat similar to the ontological argument for God. There may be some differences, but statements about black kings moving diagonally and greatest beings existing necessarily seem to be similar. What’s latin for “I am referred to, therefore I am?” That seems to be the argument in both cases.

  31. #31 Darth Dog
    October 2, 2012

    I really did like the quote that Jason included at the end of the post from the creationist math book.

    “…every once in a while we would find a person with fewer/more fingers than usual, evidence we live in a fallen universe.”

    I want to get a tshirt that says “Help, the universe has fallen and it can’t get up!”

  32. #32 RBH
    pandasthumb.org
    October 2, 2012

    I’ve always thought Wigner’s contention (and to some extent Hammings’ was over-drawn. A more appropriate title would be “The Unreasonable Effectiveness of Some Bits and Pieces of Mathematics, Many of Which Were Invented for (and Sometimes Derived from) Modeling Real Stuff, in the Natural Sciences.” :

    Apropos of very little in this discussion–at least at a superficial level–Hamming has the best line: “The Postulates of Mathematics Were Not on the Stone Tablets that Moses Brought Down from Mt. Sinai.”

  33. #33 RBH
    October 2, 2012

    (Sorry for the screwed up punctuation, though I think you can figure it out.)

  34. #34 JimV
    October 2, 2012

    Numbers are a natural, useful, and consistent concept in this universe (any intelligent species would formulate and use them, I think). I would not be surprised if they are somehow built into the neurology of our brains. Animal studies show that monkeys and many other animals can count, add, and subtract numbers over a range of about one to eight. So if someone asks me if numbers exist, I say yes, of course. There is such a thing as the concept of “two”. “Two” exists. It seems an extraordinary claim to me to say that it doesn’t, and you know what they say about extraordinary claims.

    (Other concepts exist too, but humans may never discover them, just as most animals don’t understand numbers higher than eight.)

    Speaking of two, it was pleasant to see a chance convergence of two of my favorite blogs: Bee’s comment, above.

  35. #35 Jim Harrison
    October 2, 2012

    About the unreasonable utility of mathematics: wouldn’t mathematics be more effective if the diagonal of a square were commensurate with the sides? More generally: is there some metric by which we can assess the effectiveness or lack of effectiveness of mathematics?

  36. #36 Kevin
    October 2, 2012

    The basic argument is that mathematics is so useful for describing the world solely because God, in his benevolence, designed the world to be describable in that way.

    Is that how “they” express it, or are they observing that God has created the world in such a way that we can be held morally responsible for our actions?

    It seems that materialism contradicts both parts of this observation, i.e. that the universe depends on something other than nothingness and that man has free will.

  37. #37 Charles Sullivan
    October 3, 2012

    I think the One Over Many argument is where to look concerning Number and Platonism.

    http://plato.stanford.edu/entries/platonism/#3

  38. #38 Robert
    October 3, 2012

    If I say the solution to a differential equation exists, I use the term ‘exists’ completely differently than when I say the apple on my desk exists (and will cease to exist in a couple of minutes.) So,as a few others have pointed out, the question of existence of mathematical objects really boils down to context and by what you mean with existence.

    As to the usefulness of mathematics, mathematics will always yield statements of the form: If A then B.

    Here A might simply be the axioms of number theory, or a list (physical) assumptions about an applied problem. As such, its usefulness comes in two forms.

    If you are fairly certain about your assumptions A, it allows you to make predictions B.

    If your predictions B turn out to be incorrect, you know for certain that one of your assumptions in A was wrong. This tells you something real about the system you are modeling even though the model was wrong.

  39. #39 Dan L.
    October 3, 2012

    @MattB:

    Claiming a lack of ‘meaning’ just leads us into word games. It is either an eternally true property of the universe that at some point black kings have moved diagonally in chess – or truth itself is contingent on a statement being understood by a conscious entity and/or applicable to a really existing object, and we collapse into solipsism.

    Unfortunately, word games are all we have. We’re in the position of having to formulate our thoughts imperfectly in words and then hurl the finished package over the Cartesian gulf in the hopes that someone on the other side catches it and can decipher what we meant. If I don’t understand what you mean by a statement like “numbers exist” then it’s meaningless to me, certainly. If you can’t explain it in such a way that it becomes meaningful to me it remains meaningless to me.

    Please demonstrate why the contingency of propositional truth inevitably leads to solipsism. It doesn’t seem so to me. For example, I believe that propositional truth is contingent but I am not a solipsist. Please prove that I do not exist.

    @Charles Sullivan:

    That argument doesn’t actually help very much. When Plato says “redness exists” does he mean that redness exists in the same sense that, say, the desk in front of me exists? If so, then he’s clearly wrong because I can chop the desk into kindling and it will cease to be a desk, but there’s no way to do the same thing with redness.

    Even a casual analysis of how the word “to exist” is used in everyday English demonstrates that it’s used in a wide variety of ways depending on the context. This fact is very much consistent with fictionalism/structuralism: the truth of a proposition or existence of an entity is dependent on the context in which you speak of it. It’s also consistent with the computer science interpretation of the nature of “information”. To quote a CS text on operating systems: “Information is bits plus context.” (The same bit string could be interpreted as an alphabetical character, an integer, or a floating point decimal number depending on the context.)

  40. #40 Dan L.
    October 3, 2012

    @proximity1:

    It would also help, I think, if you considered the possibility that in more than a few realms of current science practice, the currently held paradigms, when examined very carefully, show they presume a dualist formalism, a version of Plato’s forms, or of Aristotle’s naturalistic view of the physical world.

    Specific examples please.

    More than a little of standard practical science today has a few hard lessons to learn in valid reasoning and in something else philosophy takes an interest in: humility and its proper place.

    Bullshit. Philosophers are the least humble academics I’ve ever had to deal with. They presume to judge what every other field is doing right or doing wrong despite not possessing any sort of competence in those fields.

  41. #41 Anton Mates
    October 3, 2012

    eric,

    Couchloc:

    Bernays observed that when a mathematician is at work she “naively” treats the objects she is dealing with in a platonistic way

    This is not surprising, but it doesn’t really support the platonist position at all. When a baseball ump is at work, she naively treats the strike zone as a real thing too.

    Yup. In fact, Bernays’ position was closer to Jason’s than to Godel’s. He explicitly rejected the idea that all mathematical objects have some sort of mind-independent existence, which he referred to as “absolute” or “extreme” platonism. He just thought that platonist reasoning–thinking about objects “in themselves,” instead of always returning to how you could construct a physical instantiation of them–was a very productive way of building and extending mathematical models. He was arguing against pure constructivism, basically. That’s perfectly compatible with Jason’s physicalism.

    MattB,

    Unfortunately, you can’t handwave this away. If you’re going to claim the statement ‘black kings move diagonally in chess’ is only true where there are chessboards and meaningless otherwise, you’ve given ‘truth’ its own rather strange Platonic character – it has a changeable ontological status, coming into being when the rules of chess are codified and vanishing again after…what?

    There’s nothing strange about that; lots of claims can be either true or false depending on their context. “Abraham Lincoln is alive” was false in the distant past, true for a period of several decades in the 19th century, and now it’s false again. If you want to convert it into an “eternal truth,” you just have to roll the context into the claim itself. “Abraham Lincoln lived from 1809 to 1865″ is always true (in our universe, anyway.)

    “Black kings move diagonally in chess” is true if you live at a place and time where that’s an accepted rule in the game its players call “chess,” and false otherwise. If you want to convert it into an eternal truth, you have to say something like “According to the rules for chess that are almost universally accepted by players in 2012, black kings move diagonally.” That statement will always have been true. But now you’re not talking about mind-independent platonic anything.

    Not only does it appear that the universe will continue to behave in a way that is not just well-described, but described to a quasi-arbitrary degree of accuracy, by human-generated cognitive structures derived, albeit at several removes, from simple concepts of counting, but as far as we can tell it always has.

    In the first place, it’s not true that we can describe the universe to an arbitrary degree of accuracy. Current theories of physics are both incomplete and indeterministic.

    In the second place, if the universe had behaved in some mathematically incomprehensible way in the distant past–or if it was behaving that way right now, for that matter–how would we know? Ocelots don’t understand relativity, but they also don’t understand that they need to.

  42. #42 Anton Mates
    October 3, 2012

    Ha! And that’s what I get for mixing business and pleasure. Delete, Jason?

  43. #43 Jason Rosenhouse
    October 3, 2012

    Anton –

    I assume it was just the first part of the comment you wanted me to delete. Was that correct?

  44. #44 Anton Mates
    October 3, 2012

    Precisely. Thanks a lot!

  45. #45 proximity1
    October 5, 2012

    RE: Dan L. from October 3, 3:33 pm

    I’m not interested in a parlor game of “Which group is generally more arrogant: philosophers or scientists?” because my point is not to deny that philosophers, too, exhibit such behavior, nor to defend them when they do. I’m interested instead in the fact that scientists, like everyone else, have their own tendencies to be arrogant, jealous, egotistical, etc. and that the tendencies indeed influence their work in ways that touch theory formation and disputes over what is valid or invalid in theoretical controversies. That aspect of my point is, I think, beyond question.

    For an example of where a currently-held theoretical paradigm is rife with a “presume(d) a dualist formalism, a version of Plato’s forms, or of Aristotle’s naturalistic view of the physical world”–but which is overlooked by all except the minority of critics in the field who represent a theoretical challenge to the prevailing views, I offer as most outstanding, the case of molecular biology and its genetic adherents who hold to a total or near total deterministic view of the operation of genomes in the development of individuals–and, as adjunct, all that goes with this in the form of “speciation” as a conception.

    for a picture of what is wrong in prevailing genetic theory, I refer you to the principal papers and non-specialist texts of Jean-Jacques Kupiec, a molecular biologist:

    See:
    http://www.nextbio.com/b/search/author/JJ%20Kupiec

    and also his texts,

    “Ni Dieu ni gène” (2000; Editions du Seuil, Paris)

    and “L’origine de l’individual” (2008, Fayard, Paris) (English “The Origin of Individuals,” 2009, World Scientific Pubs, New Jersey, translation by Margaret Hutchings, editing by John Hutchings).

    Other interesting work on our general thinking habits is in the recent book by Daniel Kahneman, “Thinking, Fast and Slow,” (2011, Macmillan)–which deserves all the praise it has found from reviewers. Though the following isn’t his central thesis, his work tends to lend support to the view which I and others have that, when it comes to our facing a challenge to some world-organizing belief we hold as true without much or any question, we tend to immediately seek any convenient ground on which to reject (or, in the opposite case, to preserve) that view which we presume to be true (or untrue in the opposite case). This is a general tendency in all people, whatever their social, professional characteristics may be. We all tend to do this—Doctors, lawyers, judges, scientists, theologians, all of us operate in this basic manner and nothing in our make up or training makes us automatically immune to the distorting effects of the processes. Only when one makes a concerted effort to keep present in mind as often and as much as possible an awareness of this thinking-trap, this tendency, can one hope to have any significant influence over it. In general, when challenged, we immediately ask, unconsciously, (about things that we don’t believe or don’t want to believe) “_Must_ I believe that?” or, (about things which we do believe or want to believe) “_Can_ I (still) believe that?” Any convenient excuse to save us from having to accept the view we would rather reject, or having to reject the view we’d rather retain, will serve and be seized upon.

    My point is that scientists are not a special protected class when it comes to the foibles of bias and blinkered thinking. Their techniques in enquiry don’t necessarily spare them from these kinds of errors in reasoning, and, I think that over the past century, advances in science have led society to take a kind of worshipful, religious attitude toward science and, more particularly, the technology which ensues. I thnk that there is now more about science that resembles religious dogma than was the case one, two or three hundred years ago when the practitioners of the natural sciences were still making their way against a power-structure which was not of their own making. Today, the power-structure is for various intents and purposes of their making in many of the most commonly accepted ways of viewing the world—though, again, ironically, this hasn’t meant and doesn’t mean that scientists have been consistently able to avoid falling afoul of the same traps of human weakness that plagued religion-based authority and dogma centuries ago and would, again, today, if that power-structure were to regain its former hold; and that means that, today, our secular religion is technology and the science behind it–largely not understood or very vaguely understood (which suits many scientists just fine, it’s clear) by the average layman.

    I don’t really expect that the foregoing examples–or even a dozen more in addition–are going to persuade you of something you’re already predisposed to reject. And that is why this response is already more than this meager medium for communication either supports or deserves on my part If you were open to having your views influenced by challenging facts, you’ll have found in the above more than you need as a starting place. On the other hand, if all you are really interested in is the first and handiest excuse to reject what you don’t like or approve, then nothing in the above or in further examples, if I offered them, would make any difference in the outcome.

  46. #46 proximity1
    October 5, 2012

    correction : in the citation of the (original) french title of “The Origin of Individuals” ought to have been “L’Origine des individus” rather than “individuals”.

  47. #47 eric
    October 5, 2012

    My point is that scientists are not a special protected class when it comes to the foibles of bias and blinkered thinking. Their techniques in enquiry don’t necessarily spare them from these kinds of errors in reasoning,

    Of course not. That’s why scientists put so much time and effort into reproducibility. It’s why we don’t just publish results in peer review journals, we publish methods and procedures too. We want people to check our work for errors.

    advances in science have led society to take a kind of worshipful, religious attitude toward science and, more particularly, the technology which ensues.

    You understand that ‘society worships science” does not equal or provide support for “scientists are arrogant and acting religiously,” right? Your argument’s a bit of a nonsequitur. An example of a scientist acting religiously would be someone saying “bah, I’m not going to bother testing my idas. And you shouldn’t either, because they are my ideas. Just accept them! Heck, accept them even when differ from the results of yoru experiment!” Someone saying “the CDC says I should take this flu vaccine. They are scientists, so I will take it” is not an example of scientific arrogance. It is an example of classic empiricism – the speaker has confidence in science due its past history of success.

  48. #48 Dan L.
    October 5, 2012

    I don’t really expect that the foregoing examples–or even a dozen more in addition–are going to persuade you of something you’re already predisposed to reject. And that is why this response is already more than this meager medium for communication either supports or deserves on my part If you were open to having your views influenced by challenging facts, you’ll have found in the above more than you need as a starting place. On the other hand, if all you are really interested in is the first and handiest excuse to reject what you don’t like or approve, then nothing in the above or in further examples, if I offered them, would make any difference in the outcome.

    This is some of the humility you’re suggesting, I take it? Presuming to read my mind? Presuming to judge how biased I am without knowing the first thing about me? Telling me I’m not even worthy to read the sacred words you write in defense of a claim you yourself made? In reality, I completely concur with everything you say about the cognitive biases of individuals. Maybe a little humility would help you communicate with others more effectively.

    the case of molecular biology and its genetic adherents who hold to a total or near total deterministic view of the operation of genomes in the development of individuals–and, as adjunct, all that goes with this in the form of “speciation” as a conception.

    It’s not clear to me how this is an example of what you’re talking about. Could you maybe make some argument to this effect? Or do you expect me to read your mind (after all, you’ve done such a piss-poor job of reading mine)?

    My point is that scientists are not a special protected class when it comes to the foibles of bias and blinkered thinking.

    Straw man. No one is saying otherwise.

    Their techniques in enquiry don’t necessarily spare them from these kinds of errors in reasoning, and, I think that over the past century, advances in science have led society to take a kind of worshipful, religious attitude toward science and, more particularly, the technology which ensues.

    In fact, those techniques often do “spare them from these kinds of errors in reasoning”. How else could you explain the “advances in science” which have “led society take a kind of worshipful…attitude towards science”? If the techniques of scientific inquiry don’t correct for cognitive biases and errors then how is science so productive? You’d do well to read Kuhn and Lakatos on this point. Even while Kuhn was in many ways a critic of the valorization of science he nonetheless admitted that science is the best way human beings have found to arrive at reliable knowledge.

    I thnk that there is now more about science that resembles religious dogma than was the case one, two or three hundred years ago when the practitioners of the natural sciences were still making their way against a power-structure which was not of their own making

    This is just a rant. Again, try making an argument. Show me, don’t tell me. Give an example — maybe a few. Show me how science is dogmatic in such a way that the dogma isn’t corrected by more science.

  49. #49 ppnl
    October 5, 2012

    I think math is substantially different from chess in that chess is arbitrary. Take the number pi for example. A civilization on the other side of the galaxy may be computing the digits of pi now. But it is very very unlikely that they have discovered chess. It is this seeming non-arbitrary nature of math that creates the feeling of discovery over invention.

  50. #50 Dan L.
    October 5, 2012

    @ppnl:

    But pi only has the value it does in Euclidean geometry and the universe isn’t Euclidean in the first place. Let me give you an example to show what I mean: if you draw a circle on the surface of the earth, the curvature of the earth forces the diameter to be longer relative to the circumference then it would be on a “flat” circle in Euclidean space. So for any given circle drawn on the surface of the earth, the circumference is less than pi times the diameter.

    Thus the value of pi has to be derived from assumptions made about the shape of space itself. Suppose our aliens never fell into the trap of believing the world was Euclidean. Then perhaps they never would have had a reason to calculate the value of pi in Euclidean space.

    This is ignoring the idea that aliens’ thought processes could be so different from ours that their math looks nothing at all like our math. I actually think this is a possibility.

  51. #51 ppnl
    October 5, 2012

    Yes it is true that our first understanding of pi was from geometry. But its definition in modern math has nothing to do with geometry. Pi would be an important number no matter how none-Euclidean space was.

    Still you could argue that the importance of Pi is dependent on the physical laws of the universe. Could we imagine a universe in which Pi is not an important number? I just don’t think so. I think this is as silly as trying to imagine a universe where the number 3 isn’t useful.

    And no I do not think an alien math would be very different from ours. Sure there will be differences in the way it is formulated but in the end it must be a formal system of rules for manipulating symbols. Church-Turing strongly suggests that all such systems are in a deep sense equivalent.

    Not inventing Pi would be like a high tech civilization never inventing the wheel. We could go there, patent it and own them.

    Anyway I’m not saying that math “exists” out there somewhere. I’m saying that the apparent non-arbitrary nature makes it seem that way. The flip side of that is people who deny Platonism often seek to deny the non-arbitrary nature of math. I think that is a mistake.

  52. #52 Dan L.
    October 5, 2012

    But its definition in modern math has nothing to do with geometry.

    Last I checked pi is defined as the ratio of a circle’s circumference to its diameter in Euclidean space. http://en.wikipedia.org/wiki/Pi#Definition I’m open to other definitions, I just don’t know of any.

    Pi would be an important number no matter how none-Euclidean space was.

    Could you explain why more clearly? I think I’m starting to understand why this would be but I’d love to hear an explanation if you can give one.

    The Church-Turing argument is a good one; I think the incompleteness theorems are also good examples of hard limits of the capabilities of mathematics that would be discovered by any plausible alien civilization. But I’m also arguing that it’s possible that these limits don’t apply to other logical or metamathematical systems that we haven’t discovered — perhaps because we’re physiologically incapable of understanding them.

    But I’m not saying this is probable or even plausible, only possible. So ultimately we mostly agree — I also think any alien math would not be very different from ours. Although I think they’d use John H. Conway’s surreal numbers instead of the reals just because of the elegance.

  53. #53 ppnl
    October 5, 2012

    Look at one way to calculate Pi:

    Pi = 4/1 – 4/3 + 4/5 – 4/7 + 4/9 – 4/11….

    Notice that pi is not just some random number. It is highly structured. Well geometry is specifically about structure so maybe it isn’t surprising that some aspect of a simple geometry resonates with the structure of Pi.

    In a deeper sense all of math is about logical structure. So there isn’t any reason that the structure of Pi cannot resonate with aspects of pure number theory. And it does. For example the probability that two randomly chosen integers are co-prime is equal to 6/Pi^2. This connection to prime numbers connects Pi to the Riemann Zeta function and the most important unsolved problem in analytic number theory.

    And the key is that math is about structure and so cannot be arbitrary. The most important things in math are those things that are simple and yet resonate across vastly different domains of mathematics. These are the things that engage the Platonist instinct.

  54. #54 proximity1
    October 6, 2012

    RE: Dan L. Oct. 5 1:08 p.m.

    ” It’s not clear to me how this is an example of what you’re talking about. Could you maybe make some argument to this effect? Or do you expect me to read your mind (after all, you’ve done such a piss-poor job of reading mine)?”

    I have to admit that I’m not surprised that it isn’t clear to you. However, about that, it’s a “funny” thing: until I read a number of the references I’ve cited above, I’d have said the same thing you did—”It isn’t clear to me how this is an example of …” But, since I have now read a number of them , it now is clear to me how they are examples of what I’m (here in this thread) ‘talking about.”

    So, if you find that it isn’t clear to you, that’s nothing to be too concerned about; if you read some of the sources cited, it would become clear to you–or, at least, it lmight, there’s always that “danger”.

    I first came on the author (Jean-Jacques Kupiec) by sheer chance. So much in life happens by sheer chance! (Scientists make discoveries, for example.) I wasn’t looking for his work when I stumbled upon it; I had gone to the bookstore for another look at a work about Kurt Gödel which I’d seen and considered previously and found on the same visit this curious sounding text on something I’d never heard of before, “Ontophylogenesis”. Knowing nothing about it but finding it potentially very interesting, I bought a copy and took it home.

    It never occurred to me to turn to any of the others browsing nearby and ask, “Would you please summarize for me what this little text is all about?” nor did it occur to me to go on-line to a chat forum and ask that same thing of others I found there.

    Instead, I thought that if I wanted to know what the book was about, what its author had to say, then I ought to read it to discover this.

  55. #55 Sean T
    October 8, 2012

    ppnl,

    Sorry, you have to do better than that. The basic algebra, from which your formula derives, is entirely equivalent to Euclidean geometry. That was the real content of Descartes’ great mathematical advance, analytical geometry. Points are equivalent to ordered groups of real numbers, with the dimensionality of the space being equal to the number of members of the ordered group. Any theorem that can be derived in geometry has an equivalent theorem that can be derived algebraically (and vice-versa). Deriving a value of pi from algebra (or its further extension into calculus and real analysis) does not show that the value of pi is independent of Euclidean geometry.

  56. #56 ppnl
    October 8, 2012

    I never said that Pi was independent of Euclidean geometry. I said that it had mathematical structure that resonate across vastly different mathematical domains. It is not independent of any of those other domains either. It would be quickly discovered even in a universe that was not even approximately Euclidean.

    Our value of Pi would be an important number even if we lived in deeply non-Euclidean space.

  57. #57 Sean T
    October 8, 2012

    ppnl,

    Why would pi be significant in a deeply non-Euclidean space? Without a Euclidean foundation, why, for instance, would the value of the series sum 0 to infinity 4/(2n+1)*(-1)^n have any particular significance? That value would be what WE call pi, but it would NOT be the ratio of the diameter of a circle to its radius. Undoubtedly, a non-Euclidean mathematics could develop such a formula, but why would it have any particular significance? The value pi, approximately equal to 3.1415927, has significance only in Euclidean geometry. In a deeply non-Euclidean universe, the ratio of a circle’s circumference to its diameter would have a different value. There might well be formulas analogous to the one you give as an example, but these formulas would be different, and would point to the value of pi that holds in the non-Euclidean geometry.

    The problem with what you have proposed is that pretty much all of real analysis as mathematicians on earth have developed it, rests on a Euclidean foundation. That is, the formula you give is far from an example of a formula that a non-Euclidean mathematician would derive. Real analysis would be different in a non-Euclidean space, and that formula you gave would almost certainly be of little interest to the non-Euclidean mathematician.

  58. #58 Sean T
    October 8, 2012

    More succinctly: your formula (and others like it) rests on the ASSUMPTION that Euclidean geometry holds. In a universe where that’s not even approximately true, it’s unlikely that this formula would ever be derived, or that it would ever be considered significant.

  59. #59 ppnl
    October 9, 2012

    Yes it is true that in a non-Euclidean geometry the ratio of a circle’s circumference to its diameter would have a different value. It could be any random value. It could even change with position or size of the circle. But there is a way to compare and contrast all the different kinds of geometry and Pi is important for it.

    Think of an circle. Now think of a circle as a special case of an ellipse. It is a particularly simple form. Now think of an ellipse. You push past a point and an ellipse becomes a parabola. All of these are conic sections that you start to visualize in its simplest form by visualizing a circle as the cross section of a cone. You understand the general parabola by understanding the more fundamental case of the circle.

    Euclidean geometry is fundamental in the sense of the circle. It is a special case. It will be recognized as such even by inhabitants of other geometries.

    Now say you are a creature in a non-Euclidean space. Say you want to transmit information in the form of electrical signals. To counter noise you need an error detecting and correcting code. To do this you need to maximize the distance between the alphabet of your code. What do we mean by distance? This can be defined in binary by referring to hamming codes. But it turns out that there is a simpler way to visualize the problem of developing better codes. It turns out that the problem is identical to finding efficient ways to pack hyper-spheres into a high dimensional space. That is an Euclidean space. In order for you to develop your version of the internet you will need to understand Euclidean geometry. And Pi. And how your geometry relates to this special and simple form of geometry.

    Math is about structure. Structure can often be visualized as shapes. Shapes means we can use some of the tools of geometry to understand the structure even if the problem is not obviously a problem of geometry. Euclidean geometry is vital even if you don’t live in a Euclidean space. Pi is trans-universal.

  60. #60 Sean T
    October 9, 2012

    I think our primary difference here is well representative of the original topic of the thread. It seems that you regard things such as circles and cones as fundamental entities that any mathematician in any universe would discover. I tend to regard these things instead as inventions of mathematicians, invented because of their usefulness in our universe. Other mathematicians in a universe that’s distinctly different from ours would invent other structures.

    You point out that even in a non-Euclidean universe, Euclidean structures would be fundamental. I’m not saying that an advanced non-Euclidean mathematician would not come up with a Euclidean space and all it represents. However, it would likely be in a fashion similar to our non-Euclidean geometry; it would not be the fundamental mathematics of this universe. Its development would have to wait until mathematicians realized that there axiomatic systems need not correspond to physical reality, which was what had to be realized before we developed non-Euclidean geometries.

    Your claim that in a non-Euclidean universe, pi would somehow be a random value, which might be changeable depending on which circles you look at is also not necessarily true. As a counterexample, consider a non-Euclidean universe where a taxicab metric holds. If you aren’t familiar with this metric, it basically defines the distance between two points (x1,y1) and (x2,y2) as
    D = |x1-x2| + |y1-y2|. Using this metric, circles would look like Euclidean squares, rotated 45 degrees with respect to the coordinate axes. The four “corner points” of this diamond would be (-r,0), (0,r), (r,0) and (0,-r), where r is the radius of the circle. The diameter therefore is 2r. The length of each “side” of the circle is also 2r. Therefore, the circumference of the circle is 8r. In this metric, therefore, pi=4 for all circles. The value 4 would therefore have significance, not the value 3.14159. The fact that some series converge to the latter value would likely not be seen as particularly significant in this universe, at least until “non-taxicab” geometry was invented.

  61. #61 eric
    October 9, 2012

    ppnl:

    Could we imagine a universe in which Pi is not an important number?

    Well, even some people in THIS universe think we picked the wrong number to emphasize, and that the more physically useful and relevant number is pi*2.
    That would eliminate the large number of 2′s, 1/2′s etc. accompanying many uses of pi in geometry and physics.

    So, due to historical contingency, the number you currently think of as a fundamental constant might just be a factor of the real fundamental constant. ;) And it is easy to imagine a world (or species) where pi is not an important number, just as we currently live on a world where tau is not considered an important number even though physically and mathematically, it might be more important than pi.

  62. #62 ppnl
    October 9, 2012

    Sean T,

    ” The fact that some series converge to the latter value would likely not be seen as particularly significant in this universe, at least until “non-taxicab” geometry was invented. ”

    Well yes they would have to invent non-taxicab geometry. But the point is they would quickly do so. For example they would not be able to develop good error correction codes without the analytical math of Euclidean geometry. That must involve Pi. While the space they live in may not be Euclidean many other things in their world would be. At least in an analytical sense. After all we have had taxicab geometer for over a hundred years.

    So while they would likely develop math differently than us they would still see Pi as fundamental. I believe they would have their geeks calculating Pi to millions of digits just like we do.

    ” You point out that even in a non-Euclidean universe, Euclidean structures would be fundamental. I’m not saying that an advanced non-Euclidean mathematician would not come up with a Euclidean space and all it represents. However, it would likely be in a fashion similar to our non-Euclidean geometry; it would not be the fundamental mathematics of this universe. ”

    I would not say that Euclidean Geometry is the fundamental mathematics of this universe. First you are motivated because it describes one aspect of our universe – the nature of the space. There are very many more things in our universe that we use math to describe and there are very many more things that Euclidean geometry describes besides our physical space. Also our space isn’t even Euclidean anyway. Euclidean geometry is only a low energy low mass approximation. It gets weirder still when you get into quantum mechanics. I don’t think it makes any sense to talk about the “fundamental math of our universe:. All math is useful in all universes. They may just be useful in different ways. Sometimes they will surprise you and be useful in the same way despite the physical space differences.

    eric,

    The difference between Pi and Tau is just a minor point of formulation. Some other species may well think we are weird for using Pi but they would immediately recognize it as an expression of the same underlying mathematical structure.

  63. #63 Anton Mates
    October 10, 2012

    ppnl,

    Now say you are a creature in a non-Euclidean space. Say you want to transmit information in the form of electrical signals. To counter noise you need an error detecting and correcting code. To do this you need to maximize the distance between the alphabet of your code. What do we mean by distance? This can be defined in binary by referring to hamming codes. But it turns out that there is a simpler way to visualize the problem of developing better codes. It turns out that the problem is identical to finding efficient ways to pack hyper-spheres into a high dimensional space. That is an Euclidean space. In order for you to develop your version of the internet you will need to understand Euclidean geometry. And Pi. And how your geometry relates to this special and simple form of geometry.

    But the same problem comes up here: There is no reason why our hypothetical creatures have to think in terms of Hamming distance or d-error-correcting codes in the first place. Even we don’t always use that distance measure or seek such a code; whether we “should” do so or not depends on the contingent details of the communication system we’re working in, and the real-world costs of making various errors in transmission.

    Just as local space is pretty much never exactly Euclidean, maximizing the Hamming distance between your codewords is pretty much never the most optimal method of avoiding error. It’s a merely a fairly simple one–to us, with our particular brains–and often a pretty effective one–for us, with our particular communication schemes.

    Aliens might start with a non-Euclidean geometry and a “non-Hammingean” information theory, and marvel at how intimately the two theories are connected!

  64. #64 ppnl
    October 11, 2012

    ” Just as local space is pretty much never exactly Euclidean, maximizing the Hamming distance between your codewords is pretty much never the most optimal method of avoiding error. It’s a merely a fairly simple one–to us, with our particular brains–and often a pretty effective one–for us, with our particular communication schemes. ”

    Sorry dude but I think you are just playing whack-a-mole with Pi.

    Hamming distances is optimal with a large class of devices.

  65. #65 Anton Mates
    October 11, 2012

    Hamming distances is optimal with a large class of devices.

    It’s not even about the devices, it’s about the language itself and what it’s used for.

    Mistake 1: Misreading “You should kiss him” as “You should kill him”

    Mistake 2: Misreading “You should read a book” as “You should rent a book”

    Going by Hamming distance, these mistakes are considered equally severe. But if an actual person is basing their actual behavior on these messages, Mistake 1 is probably way more costly than Mistake 2.

    In communication, as in spatial geometry, the real world is always more complicated than our model of it.

  66. #66 ppnl
    October 11, 2012

    Oh man! Talk about a change of subject! The subject was math and geometry.

  67. #67 Raging Bee
    Virginia
    October 12, 2012

    Thus the value of pi has to be derived from assumptions made about the shape of space itself. Suppose our aliens never fell into the trap of believing the world was Euclidean. Then perhaps they never would have had a reason to calculate the value of pi in Euclidean space.

    No, the value of pi would be based on OBSERVATIONS about the space they occupy. Like most anti-rationalists, you are dishonestly overusing the word “assumption.”

    This is ignoring the idea that aliens’ thought processes could be so different from ours that their math looks nothing at all like our math. I actually think this is a possibility.

    Their math would reflect their material reality, just as our math reflects our material reality. If they’re in the same universe, they’ll use the same math to describe it.

    Are you a sockpuppet of Anthony McCarthy?

  68. #68 proximity1
    October 14, 2012

    couchloc @ October 2, 9:18 am

    made a very good point, I thought, in writing:

    “Bernays observed that when a mathematician is at work she “naively” treats the objects she is dealing with in a platonistic way. Every working mathematician, he says, is a platonist….. But when the mathematician is caught off duty by a philosopher who quizzes her about her ontological commitments, she is apt to shuffle her feet and withdraw to a vaguely non-platonistic position.”

    This quote seems rather fitting in this context. A good account of the whole issue of mathematical objects can be found here for those interested. Remember that Godel and Frege were platonists and this view cannot simply be laid at the feet of philosophers.

    but since then, his observation –esp. the point in bold-face–has been mainly ignored.

    More useful pre-publication reading, J.R., is in Jagjit Singh’s Great Ideas of Modern Mathematics: Their Nature and Use in chapter 10 on “Logic and Mathematics”.

    An interesting light on the key points in this discussion is to be found in Singh’s insightful survey.

  69. #69 proximity1
    October 15, 2012

    You present as an example of your frustration with « philosophy », i.e. some philosophers, this question, “Do mathematical objects exist?” and then proceed to explain how that question merits, at best, a blank uncomprehending stare from the enlightened hearer. But I think that mathematicians as well as “philosophers” have puzzled over the question, “Do mathematical objects exist?” and, further, I think that one can and perhaps ought to regard mathematics as a branch of philosophy.
    First, to give some, however slight, definition to terms, what is meant by “mathematical objects”? For me, this means, at a minimum, the so-called natural numbers, whole integers: 1, 2, 3, 4, 5, …etc. but not necessarily all related ideas that can conceivably ensue from those—for example, not “infinity”.

    So, do the natural integers “really exist”? And what does that mean, “really exist”? Is their existence, if it is “real”, that is, something that is separate from the human-kind (and perhaps other creatures) that can conceive of numbers? If no living creatures existed—as we suppose was once the case—would then numbers have existed in some meaningful sense anyway? If so, in what meaningful sense would they have existed or, if not, why not?
    I begin in this way: whole integers are at least as real as we ourselves are, and have “existed” in reality, at least as long as there have been creatures capable of conceiving of them. That long predates the existence of the game of chess, for example. So I disagree with the view that, “Mathematical objects have precisely the same kind of existence as the pieces in a game of chess.” I think that mathematical objects have a reality that is far longer than, far firmer than, and far more essential than the pieces of a chess game. Mankind got along for a very long time before the game of chess was devised or became real. Not so for numbers or the observable consequences of numbers in physical space—or, as we say today, the space-time continuum. That doesn’t, however, demonstrate or suggest that numbers existed prior to human (or other social animal) life.

    I grant that mathematics is a construction of human society and that its existence is dependent, contingent, for that reason. But I don’t think that this exhausts the question’s interesting aspects. Since human society sprang, ultimately, from what we are accustomed to refer to as inanimate matter, lifeless matter, then it follows that so did and so does everything which is contingent on that eventuality. Music, art—all artifice of human or other animal origins spring from what once evolved from inert matter. So, thought itself ultimately comes out of the evolutionary potential of inert matter.

    The implications of this are that, even if human kind never existed and never came into existence, there would, as we see reality, remain a potential for mathematics to come into existence by the eventual existence of any other evolutionary course of inert matter that comes to support cognition, an awareness of physical existence. This seems to me to presume a physical universe but not necessarily one that resembles our physical universe. Any universe in which matter as we conceive it could, I think, present all the prerequisites for a potential for mathematics simply by the existence of matter alone.

    But then, what about a universe in which there is no matter, no substance at all? As to that, I have neither an idea nor any inclination to think that this represents a meaningful question in the first place. A universe without matter is something for us is a contradiction in terms; it makes no sense to posit a universe in which there is no matter, no substance of any kind whatever. So, it seems to me that it follows that it is equally nonsensical to question the potential existence of mathematics in such a hypothetical universe.

    I think that in a universe where matter as we see it exists the potential existence of numbers and thus of mathematics is a necessary part. Ideas, thoughts, are a potential of a prior existence of matter and they have their existence, I believe, in actual material substance—which in turn is at once matter/energy in the most basic conception. That is, thought, ideas, are products of physical phenomena—the activity of brain tissue, nerves, synapses, electrochemical charges which transmit and hold sense data in cell tissue of the brain and other organs. That means that thoughts are material in nature and are transmissible as material substance to other creatures capable of sensations that can constitute thought processes. So, numbers are a necessarily potential feature of a material existence.

    What does this imply, if anything, for religious belief? I don’t think that it implies anything. There needn’t be any other ultimate basis for any thought, any ideation, mental conception, than inert matter’s existence; that inert matter might or might not eventually evolve into forms capable of mathematical awareness. But nothing super-natural is necessarily implied in that.

  70. #70 couchloc
    October 15, 2012

    For those interested there is this book on philosophy of mathematics by James Robert Brown, who is well respected. There is a chapter on platonism in mathematics as well. I haven’t read the whole thing but it may be useful (it seems much of it is available online).

    http://ondisaondi.files.wordpress.com/2011/11/philosophy_of_mathematics__a_contemporary_introduction_to_the_world_of_proofs_and_pictures_2nd_edition__routledge_contemporary_introductions_to_philos.pdf

  71. #71 proximity1
    October 16, 2012

    RE : couchloc
    October 15, 7:46 pm

    J.R. Brown P. of M. contemp intro to world of proofs and pictures –

    A fine reading recommendation and one I had not yet come across otherwise–thank you!!

    I’have had a first look at it and it’s clear that I’ll be reading that work from start to finish.

    In return, if you are able to read french, Jean-Yves Girard’s essay contained at the end of the french edition of Ernest Nagel’s and James R. Newman’s “Gödel’s proof” is also very interesting—

    Le Théorème de Gödel
    isbn; 978-2020327787

    Girard is also author of :

    Proofs and types / Jean-Yves Girard ; translated and with appendices by Paul Taylor, Yves Lafont.
    Published/Created: Cambridge [England] ; New York : Cambridge University Press, 1989.

    loc.gov link: http://lccn.loc.gov/89181884

  72. #72 Collin Brendemuehl
    October 16, 2012

    1. It’s good to see you struggling with some issues.

    2. If philosophy of science gets in the way, does not philosophy of mathematics do the same? Except that one does not, for no one can, reject all philosophy. We are systematic beings and we by nature compare our system to others. As a positivist you might enjoy Richenbach’s Rise of Philosophy of Science.

    3. “See my earlier remarks re staring blankly.” Let’s try another approach. Perhaps there are universals which are contingent. That is, values may exist apart from perception but dependent upon having been created as they are temporal. (That’s one of the challenges to Kant — the categorical imperative is treated as axiomatic/non-contingent yet requires perception to exist. Hence I agree with you re Platonism. It can be a real problem in certain areas.)