Can Atheists do Math?

by Eric Steinhart

Leibniz’s version of the cosmological argument (his Sufficient Reason Argument) runs from the continency of our universe to the existence of some necessary being. This necessary being is the ground of our universe. The ground isn’t part of our universe – it stands in no spatial, temporal, or causal relation to any thing in our universe.

The ground doesn’t even come close to matching any of the descriptions of any of the gods of mythology or even philosophy. So it’s hard to see why affirming the existence of the ground would be offensive to atheists. And yet, looking at the various responses to my last post, it seems that many atheists object. Why?

The ground is not mysterious. Leibniz has given us a great deal of information about it. First, it exists necessarily. Second, the very fact that the ground is not involved in physical relations is informative. To say that the ground is not involved in physics means that the ground is an abstract object. Abstract objects are objects that don’t participate in any spatial, temporal, or causal relations. Mathematical objects are abstract.

Mathematical objects include things like sets, numbers, vectors, functions, and so forth. If mathematical objects exist, then they aren’t physical. And, just to be clear, they aren’t concepts. Numbers aren’t things in your brain. There are infinitely many numbers; but you don’t have infinitely many things in your brain. If numbers exist, then they have an entirely objective existence that doesn’t depend on you or your thoughts at all. If you think math is all in your head, then you can’t do math.

Mathematical objects obviously play roles in science (especially in basic physics, which is intensely mathematical). The Quine-Putnam Indispensability argument says that since mathematical objects are needed for science, they exist. If you believe in quarks and gravity, you ought to believe in math. If there are no numbers, what sense would it make to use equations to describe the physical world? And, of course, you can’t just say you believe in some mathematical objects and not others. You get the whole system or none of it. And, looking back to Leibniz, mathematical objects exist necessarily.

These ideas about mathematics are known in metaphysics as platonism. Platonists affirm the objective reality of a world of mathematical entities. And, even though we can’t see or touch mathematical objects, we obviously know a lot about them. Indeed, math is the most stable and enduring part of human knowledge. Math involves proof – everything else is uncertain. Of course, platonism, like every part of philosophy, is controversial. But platonism comes with enormous benefits. Why not use them?

You can use platonism to complete Leibniz’s Sufficient Reason Argument. The following logic justifies the thesis that the ground is a mathematical object: our universe is mathematically structured; the best explanation for the mathematical structure of our universe is that it is generated by a mathematical object. It’s reasonable to believe the best explanation. Accordingly, there is some mathematical object that generates our universe. And that object is the ground.

How could the ground be a mathematical object? What does that even mean? To start to answer this question, consider a cellular automaton like the game of life. Such cellular automata can have incredibly rich physical content. Any instantaneous stage of some game of life can be encoded in a bit string – a sequence of 0s and 1s. Another way of putting this is to say that the stage supervenes on the bit string. A game of life is a series of stages; so any game of life supervenes on a series of bit strings. The bit strings aren’t arbitrary. They are generated by the iteration of a function which encodes the causal laws for the game of life. The function is Turing-computable. So every game of life supervenes on a sequence of bit strings generated by the iteration of an abstract Turing machine. The abstract Turing machine is itself just a function from numbers to numbers.

Many writers have thought hard about the possibility that our universe supervenes on the iterations of some abstract Turing machine. This is the computable universe hypothesis. (See, for instance, the work of Ed Fredkin or Jurgen Scmidhuber). I suspect our universe is too complex to supervene on the iterations of a Turing machine. But there are far more complex abstract machines. So the hypothesis that the ground is a kind of mathematical object is perfectly intelligible.

It’s easy to see why a theist might object to the mathematical ground – it directly competes with the theistic god! If the mathematical ground exists, then the theistic god doesn’t exist (or at the very least, that god is cosmologically unemployed).

Should atheists object to the mathematical ground? If so, why? Platonism gives atheists an enormously powerful metaphysics – a world of abstract, eternal, transcendental, necessary objects. But none of them are gods. And that world is knowable by reason (it’s the very peak of rationality). You’d expect atheists to embrace that.

So I’m wondering: can atheists do math?

Guest Contributor Eric Steinhart is an associate professor of philosophy at William Paterson University. Many of his papers can be found here .

About Daniel Fincke

Dr. Daniel Fincke  has his PhD in philosophy from Fordham University and spent 11 years teaching in college classrooms. He wrote his dissertation on Ethics and the philosophy of Friedrich Nietzsche. On Camels With Hammers, the careful philosophy blog he writes for a popular audience, Dan argues for atheism and develops a humanistic ethical theory he calls “Empowerment Ethics”. Dan also teaches affordable, non-matriculated, video-conferencing philosophy classes on ethics, Nietzsche, historical philosophy, and philosophy for atheists that anyone around the world can sign up for. (You can learn more about Dan’s online classes here.) Dan is an APPA  (American Philosophical Practitioners Association) certified philosophical counselor who offers philosophical advice services to help people work through the philosophical aspects of their practical problems or to work out their views on philosophical issues. (You can read examples of Dan’s advice here.) Through his blogging, his online teaching, and his philosophical advice services each, Dan specializes in helping people who have recently left a religious tradition work out their constructive answers to questions of ethics, metaphysics, the meaning of life, etc. as part of their process of radical worldview change.

  • August

    As far as I am informed, mathematics, as a set of reasoning, has nothing to do with reality. Quantum physics shows that we live not in a reasonable universe (reason being a product of brain evolution) but, rather in a nonsensical one, where particles exist everywhere at the same time, appear and disappear instantaneously and are governed not by necessity, but by chance.

    The idea that, since natural phenomena can be described sufficiently with mathematical reasoning, some metaphysical mathematical object *must* be at the heart of why reality is the way it is does not hold water.

  • http://www.ericsteinhart.com Eric Steinhart

    Let me see if I’ve got this right:
    (1) Quantum mechanics uses math;
    (2) Math has nothing to do with reality;
    (3) Therefore: Quantum mechanics has nothing to do with reality.

  • http://nojesusnopeas.blogspot.com James Sweet

    The ground is not mysterious. Leibniz has given us a great deal of information about it. First, it exists necessarily.

    Wait, wut? Listen, I don’t want to denigrate pure reason here — relativity comes to mind as a theory that was worked out by pure reason based on only a few fairly simply premises, that was later confirmed by measurement. But I think any claims to certainty based on pure logical reasoning, without empirical grounding, are inherently suspect.

    Leibniz has given us a rather convincing rational argument why the ground must exist necessarily, but this assumes the argument doesn’t contain any mistaken assumptions. I absolutely reject that you can assert that Leibniz has shown that the ground “exists necessarily.”

    I also agree with August’s second paragraph. I don’t see at all how the fact that math describes reality implies that math is reality. The map is not the territory, as they say. In fact, one can show countless examples of where a particular mathematical equation describes a physical phenomenon fairly accurately, but that the relationship turns out to be an emergent property of some more fundamental phenomenon (e.g. the rate of temperature change of a hot object left in a cold room is described by a very simple differential equation, but that differential equation is a probabilistic approximation of the effect of countless particles colliding).

    I recognize you are attempting to get at something more fundamental, but I am just not buying it.

    The contents of this webpage can be described with words. That doesn’t mean there is some “linguistic object” at the heart of this webpage — in fact if there is any “object” at the heart of it, it is a bunch of magnetic signals on a hard disk on the server(s) where it is stored.

  • http://nojesusnopeas.blogspot.com James Sweet

    Actually, on second thought, I’m going to go one step further and assert that mathematics can not be the grounding of reality. Godel showed beyond a shadow of a doubt that no mathematical system can be both consistent and complete (by showing, in a nutshell, that in any complete mathematical system one can construct a statement analogous to “This sentence is false”). And I have to believe that reality is neither inconsistent nor incomplete — right? So it can’t have a mathematical object at its heart.

    I really liked where you were going with the first post (even though my answer was that “the ground” is probably a linguistic/cognitive artifact rather than anything real), but in this post you seem to me to be in very suspect territory. It seems like you are combining an unjustified faith in logical reasoning ungrounded by empirical justification, with some serious map/territory conflation.

  • http://nojesusnopeas.blogspot.com James Sweet

    I’m also now convinced that studying modal logic leads people primarily to erroneous conclusions. ;p (I kid — mostly)

  • Brian

    “Mathematical objects obviously play roles in science…since mathematical objects are needed for science, they exist.”

    “Glider guns” are not necessary for the game of life to exist.

    “Of course, platonism, like every part of philosophy, is controversial. But platonism comes with enormous benefits. Why not use them?”

    Because it’s fatally flawed?

    Units, units be advised, we have a possible 261A reported at cammelswithhammers.com. Victim is described as an approximately 14 billion year old territory, suspect is a convoluted map of indeterminate age.

    “I don’t want to denigrate pure reason here…”

    My reasoning takes place using physical neurons.

    “Actually, on second thought, I’m going to go one step further and assert that mathematics can not be the grounding of reality. Godel showed beyond a shadow of a doubt that no mathematical system can be both consistent and complete (by showing, in a nutshell, that in any complete mathematical system one can construct a statement analogous to “This sentence is false”). And I have to believe that reality is neither inconsistent nor incomplete — right? So it can’t have a mathematical object at its heart…I’m also now convinced that studying modal logic leads people primarily to erroneous conclusions.”

  • http://aconversationwhile.com/ Kevin Scott Joiner

    We do not experience the universe. Rather, we experience an incredibly accurate simulation of the universe which we call mind. The more accurately the simulator models reality, the more accurate the predictions made about the future and the more likelihood that an individual simulator will survive and reproduce. This underlies everything from a lizards ability to know where a fly will be when it lashes out its tongue to a scientist’s ability to hypothesize and adjust theory as needed.

    Bodies of knowledge, such as empirically-backed theory, are a type of collective simulation – Chardin’s “Noosphere”, if you will. When a particular aspect is shown to be faulty (religious belief) or incomplete (Newtonian physics), an adjustment to the simulation becomes necessary. This is why we now have relativity theory, quantum theory, and string theory which, while still largely unproven, is an intriguing result of this process of simulation/prediction/testing/refinement. Math is also a result of this simulation, for which Gödel has clearly shown intractable faults and incompleteness.

    Much to the chagrin of Aristotle et al, logic and math – the underpinnings of both philosophy and science – are analogs of what is really going outside of our minds. We cannot transcend analogy; we can merely refine it. If there is a real math or logic, it is not in our minds, though our minds are a product of these.

    That said, the analogies we call math and logic are damned good replicas of what’s really going on out there!


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