When nominalists claim that math is “causally inert,” they mean that it can have no effect on the physical world.
This is obviously false.
As evidence, consider the mechanical analog computer:
Unlike their digital counterparts, mechanical analog computers calculate by way of physical components: cams, shafts, gears, pulleys, wheels, pins, grooves, balls, discs, rack-and-pinions. As a computation is executed within the machine, these interconnected parts push, pull, slide, and rotate in synchrony.
How does this constitute evidence against the nominalist’s claim that math is causally inert?
Simple: the internal operations of mechanical analog computers have nothing to do with numerals (i.e., names of numbers), and so it cannot be claimed that they operate on nominalist principles.
If nominalism is out, then its alternative must be in. That alternative is realism, the idea that mathematical entities and relationships exist as actual features of reality.
Without the objective, metaphysical existence of math as a hierarchical cause, mechanical analog computers would not be possible.
There is a lot here, so let’s try to unpack it.
While symbolic representation is core to the operation of modern digital computers, the same can’t be said of mechanical analog computers. In the digital computers we are all familiar with, patterns of binary states, or bits, “stand in” for numbers. Information is stored and manipulated representationally by particular patterns of bits, each of which is either on or off, one or zero. It is a discrete-state system that tracks unambiguous values that can be read at each step*.
In a mechanical analog computer, things are entirely different: calculations are carried out not by the algorithmic manipulation of patterns of bits that represent mathematical values, but by the physical interaction of mechanical parts. Mechanical analog computers execute mathematical calculations without quantization at all**. Instead, the metaphysically real structures of mathematics are machined into the physically real components of the computer. And it is the interactions of the parts themselves that carry out the calculations.
With gear ratios and rack-and-pinion movements, for example, there is no manipulation of the nominalist’s “man-created symbols,” just smooth, continuously variable surfaces capable of being positioned into infinite states, with no one state corresponding perfectly to any mathematical quantum or unit (except perhaps at sub-microscopic resolutions far in excess of what is needed for accurate computation–resolutions at any rate far below the “noise floor” or limits of mechanical tolerances inherent in an analog system).
Of course, the inputs and outputs of a mechanical analog computer must be labeled with the familiar digital symbols (numerals) so that human operators can interface with the device, but again, it must be stressed that mathematical quanta have nothing to do with how the analog machine carries out its calculations, nor do the mechanisms that drive the inputs and outputs snap to such quanta with perfect repeatability as they do in a digital computer, whatever the labels say. Moreover, changing the numerical labels on a mechanical analog computer would do nothing to change the inner operations of the machine, which are governed by material analogs of pure mathematical forms.
The fact that real, accurate calculations can be carried out by a machine built of nothing but analog components implies mathematical realism. It would be impossible for mechanical analog computers to exist without the prior existence of numbers–real, subject-independent things with intrinsic properties and objective relationships to each other.
Ancient realists like Plato and Aristotle were on the right track***.
Any way you look at it, the forms of mathematics must exist prior to the mechanical parts of an analog computer—indeed, in Aristotelian terms, they are the formal causes of the parts****.
They are also the final causes of the parts. Their function within the machine just is to transform inputs into mathematically reliable outputs.
Another aspect of note: The tighter the manufacturing tolerances, the better the mechanical computer. Which is right out of Plato*****. Bad mechanical computers are bad precisely to the extent that the machining of the parts fails to instantiate the metaphysically prior functions for which those parts are designed to provide an analog. A sloppily-machined cam won’t provide as accurate an output as a precision-machined cam. Implicit in such comparisons is that there exist degrees of resemblance to (or gradations of participation in) the forms. The chain of comparatives (good vs. better) must terminate in the superlative (best). Otherwise we would have no way to discern a good analog computer from a bad one.
The argument presented here is that this superlative is none other than the metaphysical reality of eternal, perfect mathematical forms.
Were math not hierarchically causal—were it not the case that the structures, functions, and relationships of mathematics exist metaphysically prior to their being instantiated as analogs into physical components: cams, shafts, gears, discs, etc.—mechanical analog computers would not be possible at all.
Conversely, if nominalism were true, it would be possible in principle to machine and assemble a mechanical analog computer based on some sui generis, human-created conceptual invention other than what we know as mathematics and have it transform inputs into reliable, consistent computational outputs.
No such computer is possible. The nominalists are simply wrong.
Analog computers show us that math is anything but causally inert. Math is the formal and final cause of the components of real machines we put to use in the real world to solve real-world problems.
Nowhere is the causal lock between the eternal, unchanging, metaphysical reality of mathematics and the contingent, physical reality we can see and touch more apparent than in the inner workings of these amazing devices.
* It should be noted that the concepts themselves are nowhere to be found in the computer. All that exist in the machine are functional abstractions (the most fundamental of which being the bit) that human beings associate with concepts. The computer itself has no concept of numbers. As computer scientist W. Daniel Hillis notes, “Naming the two signals in computer logic 0 and 1 is an example of functional abstraction. It lets us manipulate information without worrying about the details of its underlying representation.”1 Crucially, it is always the human being, both as designer and operator of the machine, who assigns meaning to the patterns of bits.
** While there can be ratcheting or stepping mechanisms that are associated with discrete values in a mechanical analog computer, the forces applied to those mechanisms are themselves continuously variable. The logical operations that activate them are functionally non-quantized.
*** Yes, there are significant differences between Plato and Aristotle. And Plato considered numbers to be not quite at the same level of existence as the forms (he is murky on this point, but the gist is that numbers are “intelligible particulars,” in Copleston’s phrase, as they are neither sensible things nor singular universals). But as defenders of realism broadly construed both Plato and Aristotle were arguing within the same tradition. In my view Plato was nearer to the mark. I think the “degrees of resemblance” argument presents a problem for Aristotelianism. If the forms just are in the objects, and if we can only know them by abstracting them from those objects, then how can we discern that one object is a better specimen of its kind than another, that one is an exemplar while another falls short? What is the explanation of the form of the formal cause of an object? I think we have to return to the “third realm” thesis of Platonic realism for answers.
**** Here I find myself invoking Aristotelianism against Aristotle, who, in his Metaphysics Book XIV, argues against the position that number exist independently as real “Ideas.” But I find his explanation flawed–a topic for a future article. I wonder what Aristotle would think of analog mechanical computers–what would he say are the formal and final causes of their parts? If not mathematical forms, then what?
***** In his Phaedo (74e), Plato has Socrates say: “Whenever someone, on seeing something, realizes that that which he now sees wants to be like some other reality but falls short and cannot be like that other since it is inferior, do we agree that the one who thinks this must have prior knowledge of that to which he says it is like, but deficiently so?” What he has in mind here is the intelligibility of the forms and our recognition that physical instantiations fall short of the perfection of those forms. Not all physical instantiations fall short in equal measure.
1. Hillis, W. Daniel, The Pattern on the Stone: the Simple Ideas that Make Computers Work. 1998. pp. 18-19.