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

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3. Against (I): The problem with idealism.

So far we have discussed (V) and (D) as background assumptions of EPLIM. Now we consider (I) (Idealism). We have seen that EPLIM is not valid without a hidden idealist premise.23 On this view, mathematical objects are viewed as constructions or ideas of the human mind, and have no existence independent of the mind. In this way it becomes immediately explicable why mathematical objects should conform to the limitations of the human mind. Each artificial object bears the mark (and imperfections) of its maker. One of the imperfections of the human mind is its inability to survey sequentially an infinite series. Consequently on the idealist view the numbers must suffer a similar imperfection: they must simply give out at the point from which surveying and construction is not possible.24 These considerations may readily be taken modus tollens as an argument for realism: there are infinite mathematical objects (such as the class of natural numbers), where these objects exceed the mind’s capacity to sequentially survey them, so such mathematical objects must exist independently of the mind.

We are going to present two strategies for refuting idealism about mathematical objects. The first strategy provides a wedge against idealism by arguing that there are real items that are not conceivable in one sense (and hence lack existence as ideas). The upshot of this discussion is that we should not insist on knowledge by acquaintance of all mathematical objects. Insofar as premise (2) of the epistemic limitations argument is a request for knowledge by acquaintance, it must be rejected. The second strategy exploits a different tack, arguing that the incompleteness of mathematical knowledge is something that should be expected if realism is true. This style of argument for realism (and against idealism) is quite indirect and less conclusive. Nonetheless, it does seem that idealism naturally predicts that there should be few (if any) limits to our knowledge in mathematics, a prediction at odds with mathematical experience.

Our wedge against idealism follows a strategy recommended by Thomas Nagel in The View from Nowhere (1986). Idealists generally equate conceivability of an item with its real existence.25 Idealists reason in accordance with principle (Id):

If X is not conceivable by us, then this fact entails that X is not real and does not exist.

In general, however, (Id) is not valid except for ideas, whose esse est concipi. It is easy enough to generate counter-examples to (Id), both mathematical and otherwise in content:

Example 1:The platypus was prima facie not conceivable by European naturalists prior to its discovery by them, but was actual and existent all along.
Example 2: The transfinite numbers were not positively conceivable to Locke and Leibniz, but if numbers are real, they are real and yet were not always conceivable.
Example 3: The infinite decimal expansion of  is still not entirely conceivable to finite minds in full detail (they cannot imagine it and survey it), but (according to realists), it enjoys full mathematical existence; there is, for example, a fact of the matter as to whether the n’th digit is 7, for all natural numbers n.
These examples are problematic. The first two examples are not obviously true if the distinction between ideal conceivability and prima facie conceivability is drawn.26 The third example begs the question against realism. In example 1, it might be true that the platypus was always ideally conceivable, even if the European naturalists failed in practice to conceive of it. Likewise, in example 2, the transfinite numbers were ideally conceivable even if Locke and Leibniz were not in a historical position to become aware of transfinite number theory. Surely had such able minds as Locke and Leibniz lived in the 20th century they would have been able to appreciate the truths of transfinite number theory.

However, appeals to ideal conceivability are problematic. It is unclear what such ideal conceivability amounts to. We have no non-circular definition of ideal conceivability. Ideal conceivability is supposed to allow us to abstract away from ‘contingent limitations’. But it is not clear where we should stop in our abstraction. Apparently everything is conceivable except that which is contradictory!27

Example (3) would also be rejected by some idealists as a tendentious example. Idealists would say that the lack of full conceivability in detail of π is an artefact of the realist way of describing and interpreting the situation. Some constructivists (intuitionists) hold that only so much of the number exists as can be computed or specified by a construction.28 The statement that ‘there is a fact of the matter as to whether the nth digit is a 7’ is an application of the law of the excluded middle, which intuitionists reject.29 Though example (3) is philosophically tendentious, it represents a mainstream view in the mathematical community.

Such counter-examples are not conclusive. So another tack is needed. Idealism implies a kind of arrogance about the scope of human understanding: idealism implies that if x is real, then x will be (ideally) conceivable by the human mind. To refute this principle, we need to find an x such that x is both real and yet not conceivable by us. Of course it is a tricky business to specify something that we cannot conceive.30 Is not to specify x to conceive x in a certain manner? One way round this impasse is to argue by analogy. In general we can see that there is most likely such an x for us, as there exist features of reality that, although evident to us, cannot be conceived by minds less powerful than our own. Nagel (1986) suggests that Gödel’s theorem is not understandable by someone with a permanent mental age of a nine year old, but still true.31 Similarly, but by analogy with more powerful minds, we cannot survey the infinitely many digits in π’s decimal expansion, but a more powerful artificial or divine intelligence may be able to do so. Moreover, despite the fact that we cannot clearly and distinctly perceive the whole expansion of π, it does not follow that π does not have a determinate identity. 32 At least it does not follow unless we have independent reasons to assume idealism.

Idealists will again appeal to the ideal conceivability of items in claiming that the conceivability of an object is necessary (and sufficient) for its reality. However, apart from the problems with ideal conceivability already noted, this move sets the standards too high. Less than ideal conceivability should not impugn the reality of an item. In fact we think there are many objects of mathematics that are inscrutable and real. A prime example is the notion of a real number. We have to allow that there are random real numbers. These real numbers correspond to random infinite sequences of digits, which comprise their decimal expansions. Random real numbers cannot be defined by rules laid down a priori and cannot be surveyed in their entirety. Their legitimacy is due to reasoning from inference to the best explanation. We know that the real number line is uncountable by Cantor’s diagonal argument. We also know that we can only compute countably many real numbers. So we infer that there are real numbers that cannot be so computed and defined. These real numbers are the `dark matter’ of analysis. They make our theory of real numbers come out right, even though we lack an individual acquaintance with each of these real numbers.33 Thus there must be mathematical objects that are not known by acquaintance.

Premise (2) of the epistemic limitations argument does not elaborate on the kind of knowledge required of the objects of mathematical knowledge. When empiricists conclude that we have no positive idea of the infinite, they presumably do so on the grounds that we lack acquaintance with the infinite.

This demand for knowledge by acquaintance is similarly present in Moore’s assumption that to make sense of statements about infinite tasks, we require ‘a kind of direct encounter with the infinite’. And it is dubious, or at least highly contentious, that we do have knowledge by acquaintance of each mathematical object; for example, it is hard to believe in recognisably distinct acts of acquaintance with a 1000 sided polygon and an 1001 sided polygon. However, this situation does not offer support for premise (1) of the epistemic limitations argument, or for the finitist conclusion. For it may well be that we have other ways of understanding infinitary mathematical statements that do not rely on acquaintance.34

The realist has an advantage in allowing that our knowledge of mathematical objects can be partial and incomplete at times.35 We do indeed have epistemic limitations that prevent us from, for example, entirely perceptually surveying an infinite decimal sequence. But such epistemic limitations do not in themselves dictate that we should therefore assume that no such sequence exists independently of our surveying and constructing it. Intuitionists explicitly take such a stance: they hold that objects do not exist until they are explicitly constructed, or at least until a recipe could be given for their construction. The realist motto is: ‘completeness of objects, incomplete knowledge’. The intuitionist (and idealist) tactic is the reverse: ‘complete knowledge, incomplete objects’.

Our claim is that realism best explains incompleteness phenomena (both of the informal sort I mentioned and the formal sort), and various forms of idealism (intuitionism, constructivism) do not. However, we recognise that this remark is not conclusive, since idealists will simply work with a different ontology (a much truncated one) that does not exhibit the relevant incompleteness phenomena. Nonetheless, classical mathematics does have phenomena (eg random real numbers) which are unknown in the restricted sense of not being objects of acquaintance (or particular construction). These phenomena can be studied using indirect methods (knowledge by description, inference to the best explanation etc.). Such incompleteness (or better, partial ignorance) is something that we should expect from a realist point of view. Realism precisely allows for facts that transcend our ability to verify (and in some cases even conceive of) them. Thus realism about mathematics suggests that premise (2) of EPLIM is false: it is not true that all the objects of mathematics are knowable and comprehensible to the human mind. The respect in which some mathematical objects are not knowable is that they are not knowable by acquaintance (‘surveyable by perception’).

Idealists should not expect such partial ignorance of all the facts about mathematical objects. In fact, in order to eliminate such ignorance, the intuitionists have to tear down classical mathematics to eliminate it. This observation about the philosophy of mathematics is actually a special case of a general point. Nagel [1986] has observed that scepticism about our knowledge of the world is to be expected (even encouraged and tolerated) on any robust realist approach to the world. As Nagel puts it: Realism makes scepticism intelligible.’36 The same holds true, mutatis mutandis, for partial ignorance (incomplete knowledge) and realism about mathematical objects. On the idealist view, any partial ignorance that we display about mathematical objects must be illusory: the objects themselves have fewer properties and less detail than we thought. On the realist view, our partial ignorance is readily explained by the richness of mathematical reality, combined with our epistemic limitations.37

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