Encyclopedia of Philosophy: Abstract objects

« to be an abstract object; but while it is not located anywhere, it has not always existed, but was devised at acertain time.

Other examples are natural languages, many if not all works of art, and words and letters in the type-as opposed to token-sense (roughly, the sense in which there are just six, not eight, distinct letters in the word'abstract') (see Type/token distinction ).

Thus while the abstract-concrete distinction undoubtedly has much to do with spatiality and temporality, it does not seem straightforwardly identifiable with the distinction between what hasspatial or temporal position and what has neither.

An alternative proposal of considerable interest is that concreteobjects are those which are, in principle, capable of being picked out ostensively, while abstract objects are thoseto which we can refer only by means of some functional expression ( Dummett 1973: ch.

14 ).

Thus we may pick out a particular tree by the words 'That beech', perhaps accompanying our utterance with a pointing gesture; but wecannot, for example, literally point to a certain shape or number - rather, we must refer to them as, say, the shapeof such and such a vase or the number of eggs in the carton ( Noonan 1976 ; Hale 1987: ch.

3 ).

3 Grounds for belief in abstract objects Many philosophers, appealing to Ockham's Razor - the principle that entities should not be multiplied beyond necessity - deem it mortally sinful to believe in abstract objects unless such belief isunavoidable, but disagree about whether it is actually avoidable.

Orthodox nominalists hope to avoid it by carryingthrough a programme of reductive paraphrase.

However, in view of the resistance of various kinds of apparentreference to/quantification over abstract objects to elimination by reductive paraphrase or re-interpretation inconcrete terms, this does not appear feasible as a completely general means of escaping commitment to abstractobjects (see Ontological commitment ).

This has led some philosophers to conclude that reference to and quantification over domains including abstract objects is indispensable to a fully adequate account of the world.There is a strong appearance that this is the case with reference to mathematical entities - numbers of variouskinds, functions and more generally, sets.

On the face of it, the natural sciences, and physics especially, requiresubstantial use of arithmetic and analysis, and the latter in turn draws fairly heavily on set theory.

This argument -known as the Quine-Putnam Indispensability Argument - provides, if accepted, a strong indirect reason for believing in numbers and sets at least: scientific theories require acceptance of mathematical theories, so that whateverreasons we have to believe that our best scientific theories are true is reason to accept mathematical theories, andso to believe in the abstract objects of which they speak.

This argument has been vigorously contested,particularly by Field ( 1980 ), who argues - in support of a new and highly unorthodox brand of nominalism - that there is, contrary to appearances, no need for mathematical theories to be true for their use in science to bejustified.

It is enough that such theories should have a certain strong kind of consistency property, which he calls'conservativeness'.

Since a nominalist can accept mathematical theories as having this property without believingthem to be true, they have no need to engage in any kind of reductive translation programme of the sort previouslymentioned - they can simply use mathematical theories while denying that they are literally true, thereby avoidingcommitment to the abstract objects their truth requires.

Among the difficulties confronting this approach, oneimportant assumption Field makes is worth highlighting.

Field takes the Quine-Putnam argument to offer the onlyground worth taking seriously for holding mathematical theories to be true, so that if he is able to undermine it,there remains no pressure to take on the ontological commitments they import.

If Field's assessment were correct,the best grounds we could have for believing maths and so on, would be indirect and a posteriori.

But thisassessment rests upon the challengeable assumption that the only statements we may justifiably accept on other-than-indirect a posteriori grounds are those directly ascertainable as true by observation.

Perhaps we should takeseriously, as he does not, the possibility that belief in the truth of mathematical statements and acceptance oftheir ontology may be warranted a priori.

4 Grounds for disbelief Unquestionably the most important arguments against abstract objects are epistemological.

One is that - in view of the presumed causal inertia of abstractobjects - to construe statements of some given kind as having their truth-conditions constituted by states ofaffairs essentially involving such objects, puts those statements irretrievably beyond the reach of our knowledge.Crudely, if mathematical statements have Platonistic truth-conditions, we could not possibly know them to be true;since we do have mathematical knowledge, Platonism is false.

In its simplest and earliest versions, this argumentrelies upon a very exacting form of causal theory of knowledge, which takes it to be an invariably necessarycondition for a thinker X to know that p, that X's true belief that p should itself be caused by, or otherwise suitably causally related to, the fact that p (see Knowledge, causal theory of ).

A problem with this argument is that while such a strong condition (just how strong depends on how precisely the vague phrase 'suitable causal relation' isunderstood) may be satisfied in standard cases of perceptual and memory knowledge, it is very hard to see how itcould be quite generally met, even when restricted in scope to ordinary empirical knowledge concerning perfectlyconcrete matters.

Our inductively grounded belief that all aardvarks have bugs is, we may suppose, causallyinduced by inspection of a large and suitably varied contingent of bug-infested aardvarks - but there is no sort ofcausal relation, however complicated or attenuated, of which it may with any plausibility be claimed both that itholds between our general belief and the fact that all past, present or future aardvarks have bugs and that itsholding is epistemically significant.

If knowledge does not demand a suitable causal link in every case, the argumentagainst Platonism collapses, at least in its present form.

A related argument alleges that no satisfactory sense canbe made of the idea that we are capable of identifying reference to or thought about abstract objects.

And onceagain, the argument in its simplest form rests upon an eminently challengeable assumption - in this case, thatidentifying reference or thought about a particular object always requires a suitable causal link between the speaker/thinker (or their utterance/thought) and the object in question (see Reference ).

Opponents of Platonism may hope to fashion more sophisticated causal analyses of knowledge and reference which are strong enough tosustain versions of these objections without being so strong as to be independently objectionable, but none has yetcome forth.

A more powerful epistemological objection appeals to the thought that, even if knowledge is not to beanalysed in specifically causal terms, we should expect to be able to provide a naturalistic explanation of ourtendency to get things right significantly more often than not, in any area where we are disposed to creditourselves with a capacity for knowledge (see Reliabilism ).

In the absence of causal or other natural relations between ourselves and abstract objects, it is hard to see how any such credible explanation might run for any. »