2. The argument from epistemic limitations.
What is striking about the classical empiricists’s arguments for finitism is the way they move from consideration of the mind’s capacities to a conclusion about the nature and existence of mathematical objects. This kind of inference (from mind to world) lends the argument an idealist character. The idealism is explicit in Berkeley’s argument for finitism:
Every particular finite extension which may possibly be an object of our thought is an idea existing only in the mind, and consequently each part thereof must be perceived. If therefore, I cannot perceive innumerable parts in any finite extension that I consider, it is certain that they are not contained in it… (Berkeley , section 124: 77-8).
Berkeley’s conclusion is that any finite extension does not, indeed cannot, contain infinitely many parts. Berkeley is a metaphysical finitist and atomist. Only extreme idealism (such as Berkeley’s) would legitimate the leap from the fact of our inability to perceive an infinity of parts to the conclusion that such parts do not exist in spatial extension.
Like Berkeley’s argument, the main thread of Hume’s argument for finitism moves from an epistemological premise to a metaphysical conclusion. Hume’s epistemological premise is that the human mind has finite powers of discrimination and can only form an idea of an interval with a finite number of parts. Hume’s metaphysical conclusion is that space and time themselves are only made up of a finite number of parts (‘minimal perceptibles’). As commonly noted, the inference from epistemological premise to metaphysical conclusion appears to depend on a kind of phenomenalism. Phenomenalism is itself a species of idealism.6 Only if space and time themselves were the product of human sense-impressions would it be legitimate to draw inferences about them based on the character of human sense-impressions and ideas.7
Idealists hold that the nature of objects (space, time, and reality itself) is itself constrained by the interpretative activity of the human mind. This reversal of priorities is the point of the ‘Copernican revolution’ in philosophy that Kant introduced, whereby ‘we try the hypothesis that objects conform to the structure of our knowledge’ (Kant (/:Bxvi-xvii, 22). Consequently, idealists should expect that they can infer the structure and properties of objects from their knowledge of the structure of human concepts.8
Idealism interacts strongly with the belief that the human mind has finite capacities, with striking repercussions for the philosophy of mathematics. Suppose that one takes the finitude and limitations of the human mind very seriously. Like a computer, the human mind has a limited ‘storage capacity’, a limited time during which it can function and apply operations, and can only take in and process a limited amount of ‘data’. If the objects of mathematics are, in some sense, (as idealists hold) constructions of the human mind, then these artefacts will necessarily conform to the limitations of that mind.9 It should be obvious where this line of thought goes: a finite mind can produce (it is said) at most a finite output, and so the objects of mathematics—the ‘output’—must be finite (and finitely many) as well.10
We can generalize the argument and arrive at (the schema of) an argument for mathematical finitism called ‘the argument from epistemic limitations’ (‘EPLIM’ for short):
(1) Human minds, being finite, are limited in their capacity to comprehend and represent objects to representing finite objects.
(2) The objects of mathematics must be comprehensible and knowable (by the human mind).
(3) Therefore, there are no bona fide infinite mathematical objects.
The conclusion (3) is a statement of metaphysical finitism, since it concerns mathematical objects. Premise (1) is a statement of a certain kind of epistemic finitism, as it concerns the human mind and its epistemic capacities. Premise (2) is intended to serve as a bridge between the two positions. Now, as it stands, the argument is incomplete. Supplementary premises are needed to derive the metaphysical conclusion from the epistemological premises. Neither premise (1) nor premise (2) is innocuous. Both premises stand in need of disambiguation and could be false. In particular, we claim that (1) is false if interpreted properly.11 More surprisingly, (2) is highly contentious, and from a realist perspective, false. As we will show, mathematical realists should tolerate the possibility of a certain amount of ignorance of the properties of mathematical objects as a consequence of their position.
To be sure, few philosophers nowadays want to be strict finitists as recommended by (3). Most would prefer to be liberal finitists. EPLIM can be recast as an argument for liberal finitism. Premise (1) of the original argument can be relaxed to allow that the human mind can comprehend and represent potential infinities. The revised argument then runs:
(1-R) The human mind can comprehend and represent only potential infinities, not actual infinities.
(2-R) The objects of mathematics must be comprehensible and knowable by the human mind.
(3-R) The objects of mathematics are at most potentially infinite, not actually infinite.
The revised argument, with its weaker conclusion, is still suspect. Like the original version of the argument, the revised argument tries to derive a metaphysical conclusion from epistemological premises. 12 From a realist perspective, such a move is illegitimate. Unless there are a priori reasons to think that objects must reflect the structure and limitations of the minds that represent them, there can be no reason to conclude anything specific about the nature of objects from our possibly inadequate conceptions of them.
It’s pretty clear that as it stands EPLIM is incomplete. It does not reflect the pivotal assumptions of idealism and epistemic finitude central to the empiricists’ arguments. The conclusion (3) simply does not follow directly from the premises without the addition of a couple of bridge principles. Many empiricists would carry these additional principles as part of their baggage and so would tend to read them into EPLIM anyway. However, we argue that EPLIM does not survive scrutiny of these bridge principles. Once these assumptions are made explicit, realists can argue that we have good reason to reject them (and the conclusion of EPLIM).