More on the nature of mathematics

The debate is heating up (in a good way) over at FQXi where I posed a question about the nature of mathematics.  While presently remaining agnostic on the issue, I want to emphasize why this is even a question at all since a number of people seem fairly wedded to the fact that mathematics is invented.  That may still be true, but there are certain aspects of mathematics that simply cannot be invented.  For instance, it is well-known that many animal species have the ability to do basic arithmetic (adding and substracting) and one of Pavlov’s experiments with dogs demonstrated that they have a fairly sophisticated ability to distinguish between certain geometrical objects.  In fact (and this doesn’t surprise me in the least), monkeys perform about as well at mental addition as college students.  Interestingly enough, what sets humans apart appears to be language.  Even though chimpanzees, for instance, can outperform humans at tracking numbers and remembering sequencing, the level of specificity possessed by humans is not possessed by any other species.  We are the only ones who assign symbols to numbers.  I find this utterly fascinating, particularly in light of my recent FQXi essay.

Thus, the questions that come to mind are two-fold.  First, perhaps some mathematics is inherent (discovered) but that some is invented.  Is the difference discrete or continuous?  Is there a gradual change from inherent to invented?  Second, is this really a question of language and, if so, how much of mathematics is a language?  It clearly can’t all be since monkeys, for example, don’t have language.  And what is it about language that can suddenly transform something from real and inherent to something that is invented and beyond the realm of possibility?  Is this just further evidence that mathematics is really a language and just a language?  If so, what do we call the logical rules of addition and subtraction that seem to be understood by many creatures?

7 thoughts on “More on the nature of mathematics

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  1. This is similar to what I posted there at FQXI:
    This isn’t about pure math as such, but addresses the issue of whether math can fully model the physical world. I say, no. For example, note that due to its being a matter of logical necessity, math cannot produce true randomness of the sort many consider manifested by quantum behavior. What I mean by “deterministic” math is that the math process can’t actually *produce* the random results. Just saying “this random variable has no specific value but can generate random numbers” etc. is “cheating” (in the sense philosophers use it), because you have to “put in the values by hand.” Such math either produces “results” which are the probability distributions – not actual sequences of results – or in actual application, the user “cheats” by using some outside source of randomness or pseudo-randomness like digits of roots. (Such sequences are themselves of course, wholly determined by the process – they just have the right mix that is not predictable to anyone not knowing what they came from. In that sense, they merely appear “random.”) I think most philosophers of the foundations of mathematics would agree with me. As for MWI as an dodge, I still ask: why doesn’t the initial beam splitter of a MZI split the wave into two worlds, thus preventing the later interference that we find?

  2. I think you’re right. And I think it is because, what we understand as math (for the most part) is a language. There are hidden “truths” underneath some of it – the natural numbers, addition and subtraction, etc. – but it is ultimately a language.

    I have no good answer for you about the MZI and MWI but then I don’t fully understand all the details of MWI.

    On the other hand, I think I have an answer for you regarding your MZI setup that you were describing. I think, if the environment interferes on a certain beam (represented by a phase shifter) between beamsplitters 1 and 2, then it ought to also do it for the same beam between beamsplitters 2 and 3. If it does so at just the right phase, the double effect seems to cancel itself out restoring the original input. This works even for a 50-50 split. But it is important to remember that first the likelihood this would happen is undoubtedly small and second even if it did happen, I’m not sure it’s anything more than a simple quantum eraser effect (not that that’s all that simple of an effect, but I think that’s what’s going on).

    1. Ian, the calculation for the output in my cascaded MZI has nothing to do with likelihood other than in the traditional sense of calculation of overall statistics for outputs from channels. Even those “statistics” is equivalent to intensity in classical physics, so the outcome is a given just with the effective equivalent of noise and variability.

      Furthermore, it is not a case of the environment meddling again at a different point, or it doing the canceling out. The recovery of the original amplitudes is automatic and happens no matter what the phase change made by the Confuser is – not just some “right phase” that is a lucky case, It is done by how the beam altered by the Confuser gets recombined later (review the explanation and the simpler version in the comment I noted.) IOW, if we start with say 70/30 split at BS1, then we get output of 30/70 (reversed from how phase changes around but same import) from BS3. It is very</i< important to appreciate, that result would be impossible if the output from BS2 would be a mixture as implied by decoherence advocates. This is genuine disproof by empirical outcome (presuming that is the outcome, and critics now are agreed on that much – just not the implications.)

      I think it would be a good idea for you to put something up about the idea here (or the other one about polarization spin, which now had a small diagram.) I wouldn't expect you to promote the validity, just put it out and see what commenters say. As for FQXi, do you know how I can apply to be a member and make posts not just comments? tx

  3. Neil,

    Hmmm. I think I’m still a little confused then. I did read your posts but I had a hard time deciphering the notation. If I get the chance, I’ll put up a diagram and a couple of equations for MZIs so you can see where I’m coming from and then comment on what would change for your setup.

    As for FQXi, to become a member you have to either a) get a grant from them, b) win first, second, or third prize in one of their essay contests, or c) be nominated by more than one present member. I became a member via option b).

  4. Hello Ian,

    Mathematics is both a logical system and a language. The facts (logical deductions) of mathematics are discovered, but the formalism (language) is invented. I think most of the paradoxes and philosophical riddles regarding mathematics disappear when one keeps in mind the fact that we use one word (mathematics) to refer to multiple things (the logical system, its results, the formal language, the act of computation, the college major, and on and on). It’s similar to quantum mechanics. There’s a big difference between the energetic state of an atom and the scribbles on a piece of paper that represent that state, but both are referred to as quantum mechanics.

    It reminds me of an argument I used to have with my brother: He said, no one can ever make a grammatical error, because if it’s an error, it’s not grammatical. Again, if you accept the fact that ‘grammatical’ means more than one thing (‘having to do with grammar’ and ‘correct use of grammar’) the disagreement disappears.

    I’ve found that most philosophical debates (including the question of the invention/discovery of mathematics) follow this same pattern. We can always come up with paradoxes if we allow a little bit of slip in our definitions. Natural languages are pretty bad for logical analysis because every word has many meanings and the differences between them can be very subtle. Another example – Descartes proves that the mind is separable from the body:

    1 – I can imagine my mind existing separately from my body.
    2 – Anything that can be coherently imagined must be possible.
    Thus it is possible to separate the mind from the body. QED

    But was it really his mind that he was imagining as being separate from his body, or something qualitatively similar to his mind, but not identical? Is it his ‘real’ mind or only his own mental representation of his mind? These are fairly subtle differences that don’t usually become crucial until we start using the single word ‘mind’ in a deductive proof. Also, what does he mean by ‘possible’? Logically possible or physically possible? He seems to switch meanings between premise and conclusion. This imprecision in natural language is what motivates the development of artificial languages like mathematics which are so useful for physics and other sciences: Every ‘word’ is precisely defined so there is no ambiguity in the meaning of a statement, and accurate deduction is possible. In other words the invented language of mathematics is implemented to discover the facts of mathematics.

    I would disagree with the statement that monkeys (and animals in general) don’t have language. They may not have written language, but that doesn’t mean they don’t think symbolically, and they quite obviously communicate with each other. There is lots of evidence that many animals even have some form of spoken language, and if you observe them carefully enough you can figure out what some of the words mean and speak them back to them. I can make my cats look up at the trees by ‘saying’ their word for ‘bird’. Some monkeys have distinct words for ‘snake’ and ‘eagle’. Say one and they climb a tree. Say the other and they hide under a bush. On a more mundane level, growls, hisses, chest-beating, whimpering, and so forth all constitute what can easily be described as symbolic language. When we insist on using ‘language’ to mean ‘human language’ or ‘written language’ that we come to questionable conclusions like ‘animals don’t think symbolically.’

    I’m sure I’m not saying anything you haven’t heard before, but as a person with degrees in physics, mathematics, and philosophy, I look forward to any opportunity to discuss all three at the same time. I am now an academic geophysicist, and most of my colleagues shun philosophy as useless and non-rigorous (I think they’re right, but it’s still fun to talk about). I could go on and on (I haven’t even touched on the physics/mathematics topics you mentioned in an earlier post, or the resolution of Zeno’s paradox, which is obvious to freshman physics majors, even though I’ve seen it presented as unresolved in graduate philosophy courses), but this comment is long enough. Thanks for indulging me.

  5. Joseph,

    Thanks for the great reply and my apologies for not responding sooner. I have been very, very busy (too busy) and haven’t had time to reply (let alone put a new post up). But I like a lot of what you say.

    I do agree that some animals have a “language.” In fact I’ve heard some reports of recent research that has even managed to “translate” this language in some animals. So monkeys were a poor example. Perhaps ants would be a better example. Or amoebas. But then we start getting dangerously close to a discussion of emergence.

    So, is mathematics an emergent phenomenon?

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