What Finite Minds Can Know: Epistemic Limitations and the Existence of Infinite Mathematical Objects

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What Finite Minds Can Know: Epistemic Limitations and the Existence of Infinite Mathematical Objects

The classical British empiricists were finitists. They argued that we lack an adequate idea of any infinite object on the grounds that we cannot perceive an infinite object. The empiricist approach lingers in contemporary constructivist philosophy of mathematics, such as the views of A.W. Moore (1989), (1990), and (2007). Mainstream contemporary mathematics is undeniably infinitist: one needs infinite sets to do real analysis and set theory. This suggests that the main philosophical motivations for finitism are flawed.

We reconstruct what we take to be the main argument for finitism: the argument from epistemic limitations (‘EPLIM’). We identify several key epistemological assumptions—verificationism, idealism, and the finitude of perceptual experience—needed to make the argument valid. We indicate why we hold that all these assumptions are false. We consider in detail some arguments against idealism about mathematical objects. We conclude by developing a realist response to EPLIM and showing how epistemic limitations are fully compatible with the reality of infinite mathematical objects.

  1. Introduction

Several branches of mathematics—such as number theory, geometry, topology, real analysis, and set theory—express truths that can only be satisfied by infinite domains. From a realist perspective, mathematical knowledge thereby requires a vast ontology. How can this vastness be reconciled with the apparently limited epistemic capacities of human agents? The human mind has finite computational and perceptual resources. The infinite objects of mathematics threaten to outstrip the mind’s ability to know them. This problem, dubbed ‘the problem of remoteness’ by Lavine [1994], is essentially an epistemological problem.1 Like most epistemological problems, the problem is the result of bad epistemology, an artefact of the way the problem is formulated. Or so we shall argue.

Mathematical knowledge provides a special test case for empiricism.

Roughly speaking, empiricists hold that the source and basis of all meaningful concepts is sensory experience. Mathematics appears to provide knowledge of infinite objects (or at least of the concept of infinite objects). The concept of infinite object does not seem to be derivable purely on the basis of sensory experience. All that sensory experience presents is a finite object (a finite line segment, a finite collection of objects) together with the open ended possibility of repeating operations (such as dividing and counting) indefinitely. Thus the limits of our experience together with the infinite character of some mathematical knowledge put severe pressure on empiricist epistemology. Moore [1990] suggests that the existence of the infinite is ultimately empiricism’s undoing:

There can be no doubt that empiricism was one of the great philosophical movements, of deep and lasting significance: but this is partly because of lessons we can learn from its ultimate failure. It is a very important feature of the infinite that it helped signal that failure (Moore [1990]: 83).
Before we can evaluate Moore’s insightful comment, some terminological preliminaries are in order. We can distinguish between three general varieties of finitism: metaphysical, semantic, and epistemological.2 The tripartite division between kinds of finitism is as follows:

  1. Metaphysical finitism—the belief that there are no infinite objects.

(ia) Mathematical finitism—the belief that there are no infinite sets or infinite numbers.

(ib) Physical finitism—the belief that the universe (or space, time, is finite in size, or contains finitely many members.

  1. Semantic finitism—the belief that talk about the infinite is non-sense.

  2. Epistemological finitism—the belief that only finitary reasoning is epistemically secure; reasoning involving an infinite number of steps or the use of quantifiers ranging over infinite domains is suspect.

With regard to (ia) mathematical finitism, we can further distinguish between strict and liberal varieties. Strict mathematical finitists believe that there are upper finite bounds to numbers and other mathematical entities.3 Liberal mathematical finitists think that there may be a potential infinity of numbers (or other mathematical entities), but deny that there is an actual completed infinity of numbers (or other mathematical entities). In mainstream, classical contemporary mathematics, strict mathematical finitism is assumed to be false. Infinitism has won the day, as indicated by the widespread assumption of the Axiom of Infinity.4 Moore’s comment suggests an argument from the truth of infinitism to the falsity of empiricism. This inference raises several philosophical questions:

  1. Does empiricism compel finitism?

  2. Are there any good reasons to be finitist? How about epistemological considerations?

  3. Does the practice of classical mathematics provide an argument against finitism?

Old fashioned empiricism is not a live option. However, several positions in the contemporary philosophy of mathematics (e.g Dummett [1966], Moore [1989], Wright [1982], [1985]) share empiricism’s view that mathematical knowledge must be grounded in the appropriate experience of mathematical objects. Moore [1989] argues that we cannot make sense of the idea of experiencing an infinite object and therefore that statements about actually infinite objects are in fact unintelligible. However, Moore allows that we can make sense of statements about potential infinites. Moore thus argues for liberal semantic finitism.

The classic example of a liberal finitist position is Gauss’s interpretation of the limit concept. Gauss stated:

I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler, the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction (Gauss [1831]:216)

Liberal finitism appears to present the best of all worlds. On the one hand, it allows for the use of the infinite in the calculus and elsewhere. On the other hand, the actual infinite—which is epistemologically suspect—is banned from mathematics. However, even liberal finitism is in tension with the mathematical practice of quantifying over infinite sets. Using Quinean methodology, such quantification carries an ontological commitment to completed, infinite objects.5

We argue that finitism (both strict and liberal mathematical varieties) is not justified by traditional epistemological considerations. Section §2 of this essay offers a reconstruction of the argument from human epistemic limitations used by finitists. (This argument is referred to as ‘EPLIM’). Section 2.1 shows that Hume and Berkeley embraced a version of the argument from epistemic limitations. Section 2.2 excavates some of the bridging principles needed to make the argument valid. Section 2.3 attempts to refute one such bridging principle, the alleged datum (D) that human beings cannot encounter an infinite object. Section §3 argues against bridging principle (I) which depends on a consequence of idealism. Section §4 presents the realist response to the EPLIM argument. We conclude in section §5 that the vastness of mathematical ontology coheres well with due acknowledgement of the mind’s epistemic limitations.

To be sure, there may be better arguments for finitism than the argument from epistemic limitations presented here. Nonetheless, as we show, the argument has been tremendously influential. Finitists from Hume to such ‘Wittgensteinian’ finitists as A.W. Moore have embraced some version of EPLIM. If we are right, then the single most influential philosophical path to finitism has been closed off. Infinitists can have their cake and eat it too: they can maintain the existence of the actual infinite in mathematics and respect human epistemic limitations.

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