It’s Hard Out Here for a Platonist

Alex Knapp took issue with my quasi-Platonism (and possible math idolatry) over at his tech blog for Forbes.  Here’s the crux of his argument:

The bottom line is that human beings have brains capable of counting to high numbers and manipulating them, so we use mathematics as a useful tool to describe the world around us. But numbers and math themselves are no more real than the color blue – ‘blue’ is just what we tag a certain wavelength of light because of the way we perceive that wavelength. An alien intelligence that is blind has no use for the color blue. It might learn about light and the wavelengths of light and translate those concepts completely differently than we do.

In the same way, since the only truly good mathematicians among the animals are ourselves, we assume that if we encounter other systems of intelligence that they’ll have the same concepts of math was we do. But there’s no evidence to base that assumption on. For all we know, there are much easier ways to describe physics than through complicated systems of equations, but our minds may not be capable of symbolically interpreting the world in a way that allows us to use those tools, any more than we’re capable of a tool that requires the use of a prehensile tail.

It’s a good attack on my metaphysics of math, but I think this line of attack takes a lot of other abstract ideas down with it.  In fact, it’s so broadly applicable that it raises the possibility that this line of attack is self-refuting, depending on how confident we are that parts of our subjective experience count as ‘real.’

Let’s start by expanding Knapp’s point about ‘blue’ being a human-dependent label for an arbitrary span of light wavelengths.  If we’re trying to go to the absolute base level of ‘real’ we can go a lot further.  Quantum wave functions are the most fundamental we’ve gotten, and you can make the argument that they’re what’s what’s really real and that humans are just an artifact of scale, of putting arbitrary labels on phenomena too big for us to understand.

That’s the endpoint of this argument, but you’re probably more familiar with it’s application to moral judgement.  Talking about The Good is a labeling error.  We may be able to say some things seem more or less good, but it’s all continuous and who’s to say we’re looking at the whole thing with the right metric, anyway?  Maybe being SuperHappy is the most babyeating thing you can be.  [See "Three Worlds Collide" a short story by Yudkowsky if that last sentence didn't make sense].

Knapp’s argument isn’t excluding math from the category of Forms, it’s denying the existence of any Forms at all.  I think that’s too much epistemological modesty.  At the very least, the evidence for math is compatible with my explanation as well as Knapp’s.  He thinks the correspondence between abstract math and physical phenomena shows that we generalized from our experience.  I think it means we get a sense of something we can’t perceive directly by seeing it imbued in the physical world.  We see it by its shadow.

You can accuse me of multiplying entities past what is required for an explanation, and you may be right.  But if you think that there’s some kind of external metric for judging our actions, that we have imperfect, indirect access to, and isn’t something that necessarily corresponds to dominant game theory/evolutionary strategies, you’re going to have to come up with a criteria that specifically disqualifies math from fitting into the Form-like category you’ve reserved for morality.  If you’re comfortable jettisoning all of these transcendentals, that’s going to be a much longer and harder fight.

About Leah Libresco

Leah Anthony Libresco graduated from Yale in 2011. She works as an Editorial Assistant at The American Conservative by day, and by night writes for Patheos about theology, philosophy, and math at www.patheos.com/blogs/unequallyyoked. She was received into the Catholic Church in November 2012."

  • Lukas

    Can’t you just define math as the set (or maybe, as subset) of the necessary relationships between concepts?

    You write, “He thinks the correspondence between abstract math and physical phenomena shows that we generalized from our experience. I think it means we get a sense of something we can’t perceive directly by seeing it imbued in the physical world. We see it by its shadow.” It sounds to me like this is more of a question about physics than about math… it’s a question of why math is so good at modeling the physical world. That might be a different question from the question of whether numbers are Forms.

  • Jonathan

    Actually, the wave function is something of a mathematical construct for dealing with quantum mechanics. Strip away one level of abstraction, and you get the state vector. The wave function is just a state vector in the configuration space basis. But the state vector is also a construct because it includes non-physical information, such as magnitude and overall phase. It also can only express “pure states”. All other states, “mixed states”, are statistical ensembles of pure states and must be expressed using a density operator. But the density operator is in turn just a mathematical construct because it can express impossible states, such as states that don’t have exchange symmetry, or have the wrong exchange symmetry class.

    • http://blogs.forbes.com/alexknapp Alex Knapp

      Actually, there was a paper that just came out on ArXiv indicating that wave functions actually exist beyond their mathematics – I’m interested to see if it survives peer review.

      • Jonathan

        I’ve seen this, but haven’t had time to read it. But it seems to be saying that the pure state, the ray in space of state vectors, is the one with physical meaning. The wave function would still be a mathematical way of considering the state.

  • Tom

    It’s hard to find substantial arguments in Mr. Knapp’s post. The analogy with blue is particularly troubling, since many philosophers would acknowledge that colors and other ‘secondary qualities’ are indeed mind-dependent. One wonders whether Mr. Knapp would deny, e.g., that the property of being three-dimensional, or the property of having mass, is real.

    In any case, there are very serious problems with both conceptualism (‘they’re mind-dependent’) and nominalism (‘they don’t exist’) about numbers.

    If conceptualism were true, then it was false that there were eight planets in the solar system before humans existed. But of course it’s more reasonable to say that there were indeed eight planets in the solar system before humans existed, and that four was greater than three even before humans existed.

    If nominalism were true, there would be no explanation or truthmaker for the proposition that, e.g., the number seven is prime. No doubt, Mr. Knapp et al. would say that we’re really just talking about a concept: seven. But concepts aren’t prime; they aren’t divisible by anything. And this once again inherits the problems of conceptualism; the number seven would be prime even if no one had ever counted that high.

    The most common anti-platonistic arguments are Ockhamistic in character: that adding numbers to our ontology would multiply entities beyond necessity. But clearly numbers add to our explanatory power. Others argue that we should assume that an ontology will be smaller until it is proven bigger. But no one’s ever really given an argument for this; why should we not expect the world to contain more entities and entity-types?

    • deiseach

      As an Old Solarist, I maintain that there were and continue to be nine planets in the Solar system. Who am I to insult the Mi-Go, the dread Fungi from Yuggoth? :-)

      But that’s a perfect example; the definition of “what constitutes a planet?” is a human-derived, mind-dependent concept about which we (and the International Astronomical Union) can argue, but the numbers eight and nine existed independently of human definition.

      We could call them “itlle” and “baroop”, but the fact remains that “baroop” is greater than “itlle” and humans have “baroop” fingers and “ketcha” thumbs, for a total of “meeksax” digits.

      • deiseach

        Well, some humans (who suffer from polydactyly) have baroop or even more fingers; the rest of us have itlle.

        That’s what happens when the mathematically-challenged post late at night.

      • Tom

        Right; that’s a good example.

        Another is the riddle, ‘If a donkey’s tail were called a “leg,” how many legs would a donkey have?’ The answer is ‘four.’ Even if human beings had never decided to invent a word, the reference of which was a leg, legs would still exist.

    • http://blogs.forbes.com/alexknapp Alex Knapp

      I’m not a conceptualist. My approach to math is a hybrid of embodied mind & empiricist schools.

      My point is simply that there’s no reason to assume Forms. Mathematics is a convenient way for humans to perceive and understand the universe, as is color. But the fact remains that the color blue is merely the human tag for “photons travelling at a wavelength of about 475 nm.” We use colors the way we do because we have eyes that detect light within a particular spectrum. A species with no eyes, or eyes that only see in, say, infared, would never develop the concept of color. The way they’d perceive and interact with light in the visible spectrum would be the same way we deal with X-ray emissions – we distort the image so that it’s visible to us.

      In the same way, math is how we, as humans, model the physical world. Because our brains are capable of thinking mathematically. But it may well be that alien intelligences have different ways of modeling the physical world that don’t use math as we perceive and understand it.

      That 2+2=4 is a construct based on the way human beings think and perceive the world around them. It doesn’t exist in some mystical fairyland – it works the way it does because our brains and senses work the way they do. Every way in which we observe, interact, and theorize about the world is inevitably filtered through not only the limitations of our senses, but also our cognitive structures.

      It may well be that mathematics is a universal means of interpreting physical phenomena for intelligent beings capable of a certain level of language and technological capability. But right now the only dataset we have is an n=1 –> it’s not enough to draw a hypothesis out of.

      As it stands, math is a very useful and practical means for humans to generalize from specific perceptions of different phenomena. There’s nothing, however, to suggest that numbers exist in some ethereal plane and peek their head into dimly lit shadow for us to see.

      • deiseach

        I still have to disagree here: “blue is merely the human tag for “photons travelling at a wavelength of about 475 nm.”

        Nevertheless, the photons are travelling at that wavelength and we do have a sensory apparatus that perceives their effect as colour.

        Suppose before we had invented the technology to measure infrared radiation, let us say Sir John Mandeville’s traveller’s tales were true and the first explorers to Ind or Prester John’s Land met an intelligent species that evolved along the lines of reptiles (imagine bipedal pit vipers).

        We cannot “see” infrared the way they can; they cannot “see” blue the way we can. Now the snakemen could call us lying apes, and we could call them lying serpents, but the fact remains that even if we cannot perceive each other’s concepts, the photons are still out there travelling at their respective wavelengths. Denying the objective existence of blue or ssasssness (what the snakemen call the stimulus they perceive in the infrared) because they are concepts formulated in our differing sensory appartuses does not mean that the photons don’t exist.

        Same way: we can say that 2+2=4 is just a human way of dealing with ‘this pile of things and that pile of things mean this town owes the king that much tax’, but that does not mean that there is not an objective concept out there that is described in our limited way by “two” and “four” (and “equals”). Otherwise, how come ten fingers today is ten fingers tomorrow, and not eleven fingers, or nine, or ten in the morning and eleven in the evening?

        • http://blogs.forbes.com/alexknapp Alex Knapp

          We cannot “see” infrared the way they can; they cannot “see” blue the way we can. Now the snakemen could call us lying apes, and we could call them lying serpents, but the fact remains that even if we cannot perceive each other’s concepts, the photons are still out there travelling at their respective wavelengths. Denying the objective existence of blue or ssasssness (what the snakemen call the stimulus they perceive in the infrared) because they are concepts formulated in our differing sensory appartuses does not mean that the photons don’t exist.

          I don’t disagree! I’m saying that ‘blue’ is how we perceive photons – its entangled in our own sensory apparatus. There’s no Platonic idea of ‘blueness’ that exists outside of the entanglement of human cognition and photons with a wavelength of 475 nm.

          • http://rhett.weatherlight.com Rhett Aultman

            I think it’s also worth noting that “photons with a wavelength of 475 nm” is itself a description and model which, at the end of the day, distills down to things that our senses perceive. Without the means of it becoming apparent to the senses, it doesn’t get recognized or described. When it becomes apparent, it often gets an description based on how it was initially recognized. This is why we have the awful word “electromagnetism,” as the force was first perceived through separate and apparently unconnected means and only later connected together.

        • http://rhett.weatherlight.com Rhett Aultman

          Otherwise, how come ten fingers today is ten fingers tomorrow, and not eleven fingers, or nine, or ten in the morning and eleven in the evening?

          Technically, we can’t have complete certainty about that, just as you can’t have complete certainty that the sun will rise tomorrow. You can, however, find reasons why it’s been that way every day before, and take certain futures as incredibly likely.

          Why is ten X today the same as ten X tomorrow? Because it’s very hard to change those definitions that quickly. We have, however, evolved the concept of what 10 is over time. The numbers qualitatively change with the evolution of math.

  • deiseach

    “An alien intelligence that is blind has no use for the color blue.”

    Don’t even have to go to alien intelligences; there are blind humans. Doesn’t make “blue” not-real if organism X can’t perceive it. Humans can’t perceive infra-red radiation the way a pit viper does, but that does not make it “unreal” for us, and we’ve developed technology to exploit it.

    Second point: I think that xkcd cartoon would be even more awesome if the line “You’re a brain” was instead “You’re a mind”.

  • Alex

    Here is a much more substantial exposition of the issues surrounding platonism and formalism.

    http://www.dpmms.cam.ac.uk/~wtg10/philosophy.html

    I also have one question regarding platonism: if mathematical propositions express truths then is the parallel postulate true or false? Or is does my statement expose a misunderstanding

    “Quantum wave functions are the most fundamental we’ve gotten, and you can make the argument that they’re what’s what’s really real and that humans are just an artifact of scale, of putting arbitrary labels on phenomena too big for us to understand.”

    I must admit that I have no idea what this argument has to do with Platonism however I don’t agree with it at all. The wave function is part of a physical model that was created in order to explain phenomenon that classical mechanics fails to explain. In describing the macroscopic world it is no more accurate or “fundamentally correct” than Classical mechanics.

    “it’s denying the existence of any Forms at all”

    Sounds good to me. I haven’t heard any convincing arguments for an “external metric” for judging actions. The closest I’ve heard of is Natural Law morality.

  • http://barefootbum.blogspot.com Larry, The Barefoot Bum

    Again, I’m not sure where you’re going here. I don’t think there’s any “external [moral] metric”, so I don’t see any problem jettisoning the independent existence of any so-called transcendentals.

    I hold that an empiricist epistemology/ontology has its “foundation” in phenomenology. I put “foundation” in scare quotes; it’s a real foundation in the sense that phenomenological statements are accepted as true outside of the epistemological method, but we don’t derive complex epistemological statements from this foundation. Rather, we construct a theoretical ontology from which we can derive our phenomenological foundation. This method does not give us deductive certainty for our ontology, but it appears to be far more powerful than trying to actually construct an ontology from an axiomatic foundation, and it requires us really to bite only the one philosophical bullet (lack of metaphysical/ontological certainty).

    Also note too that “blue” does not, strictly speaking, label a wavelength of light. That there’s such a thing as “light”, which has “wavelength”, is an ontological theory. I think it’s much better to say that “blue” labels a particular foundational subjective experience. It follows then (although the argument is more complicated than I have time for right now) that the idea that there are other minds, which have similar subjective experiences, and with whom we negotiate a shared vocalization/inscription we use to label the common experience, is itself a scientific theory we use to reconcile the phenomena (subjective experience) both of experiencing colors as well as observing how other people behave.

    The problem of “transcendentals” disappears in phenomenological empiricism. So-called transcendentals are just very low-level, general, abstract hypothetical elements of a theory of the world. They are not independently justified, they are justified to the extent that they help us create a compact and efficient ontology to account for our foundational phenomenal experiences.

    • http://blogs.forbes.com/alexknapp Alex Knapp

      Yes, this.

    • http://rhett.weatherlight.com Rhett Aultman

      Bravo.

  • Daniel A. Duran

    There are problems with what he saying:
    “But numbers and math themselves are no more real than the color blue – ‘blue’ is just what we tag a certain wavelength of light because of the way we perceive that wavelength.”
    If numbers are not real, what are they? Is 2+2=4 a contingent and mind dependent proposition?

    Saying that 2+2=4 is contingent is incoherent. Otherwise he’s ceding the point to Leah that numbers exist logically prior to the existence of minds.

    • Patrick

      Not necessarily. He could think that numbers are a label for the relationship between objects. In that case numbers could be neither contingent nor mind dependent. If believing that things have relationships regardless of human minds is platonism, then sign me up… but in my experience, platonists believe more than that.

      • Daniel A. Duran

        “He could think that numbers are a label for the relationship between objects.”
        He can try that but that would push things only one step further, is math dependent on the existence of objects?

        • Patrick

          No. The relationships between objects are ideas. Ideas aren’t dependent on the existence of the things they describe.

          • Daniel A. Duran

            “No. The relationships between objects are ideas. *Ideas aren’t dependent on the existence of the things they describe.”*

            How does that help his position that ideas (or mathematics) are not real?

          • Alex

            What do you mean by real?

  • deiseach

    Speaking of numbers, I saw this and immediately thought of you, Leah :-)

    The artist is called Kumi Yamashita, the piece is called “City View, 2003″ and it’s all done with light striking the material obliquely and creating the shadow.

    • leahlibresco

      Oh, I love this.

  • Quid est veritas

    Are we mixing Math and Philosophy? And why did he bring up aliens?

    • deiseach

      Because everything is better with aliens.

      And bacon.

      :-)

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