NATURAL NUMBERS AS TROPES

TROPICAL NUMBERS

Claudio Costa/UFRN/Brasil

e-mail: ruvstof@gmail.com

 

Summary

A natural number can only be a trope if there is a way to understand it as some kind of spatiotemporally localizable property. In this paper, I develop a strategy to explain applied natural numbers as having this property. I also show how this strategy can be extended to what is common to all equinumerous applied numbers, namely, to abstractly considered natural numbers.

Key-words: trope theory, natural number, ontology, philosophy of mathematics

 

Trope theory aims to give an ontological account of the whole world based on empirical building blocks called tropes. I characterize a trope as a spatiotemporally localizable property, notwithstanding its vagueness. This characterization is easily applicable to properties like qualities and forms, external or internal, simple or complex, the last ones being homogeneous or heterogeneous. It is also easily applicable to relational properties, insofar as they are spatiotemporally localizable. Material objects are bundles of compresent tropes. Even very diffuse things like the electromagnetic field of the Earth or the ThirtyYears War should be understood in tropical terms since they can be seen as made up of very diffusely located spatiotemporal properties.

  A vulnerable point of trope theory is that it seems unable to explain the so-called abstract entities like those of mathematics. Against this limitation, my aim in this paper is to sketch a strategy to explain natural numbers as tropes. This can be of obvious interest since natural numbers are touchstones of mathematics.

I

A preliminary point concerns the usual strategy to construct universals as sets or sums of strictly similar tropes (Williams 1953 I: 9; Campbell 1981). The universal of redness, for instance, is the set of all tropes of red that are strictly similar one another. However, this answer has well-known shortcomings. First, some consider sets to be abstract entities, universals in a Platonic or some Aristotelian sub-Platonic sense. Second, there is a well-known problem concerning our treatment of strict similarities: Either they are instances of the universal of strict similarity, contaminating trope-theory with realism on universals, or they are also tropes. If they are tropes, however, then two strict similarities between tropes must be strictly similar one another, what commits us to conceding a second-order strict similarity. But then we would soon need a third-order strict similarity of second-order ones, falling into an infinite regress. A further problem is that sets can change the number of their elements, whichmeans that sets of strictlysimilar tropes could easily change their size, while universals cannot change their size since they do not have sizes. An appeal to open sets would not be helpful since although they exist in our minds and as written symbols, it is doubtful if they exist at the ontological level.

  My way to deal with the problem is free from these burdens. It is simpler and inspired in the particularism of English empiricism. Instead of appealing to sets, I suggest that to build tropical universals we proceed like a geometer building a proof by means of an example drawn on a blackboard: the example serves as a model for all cases (Berkeley 1710, Intro., sec. 12). Doing so, all that we need in order to construct a universal is just one trope that we arbitrarily choose to use as a model. Then we define the universal as either this model or any other trope that is strictlysimilar to this model.That is, calling T*the trope T used as a model, we can define the universal as follows:

Universal for a trope T (Df.): T* or any other trope strictly similar to T*.

Giving a simple example, suppose I am acquainted with a patch of vermillion of cinnabar, a very specific color. Now, suppose I intend to use this trope as the model-trope Tv*. This allows me to build the universal of this color as the trope Tv* or any other trope strictly similar to Tv*. Certainly, Tv* in practice does not need to remain the same. You can choose any other strictly similar patch as a model. That is, the models can be randomly chosen. In this way, we are free from the urge to build higher-order strict similarities, since we may consider any chosen trope T in terms of its first-order strict similarity or lack of strict similarity with Tv*. Moreover, there is no question about size in this operational view of universals,since we dropped the appeal to sets.

II

Coming to our attempt to build numbers as tropes, empiricist accounts can give us a first clue. Locke, for example, saw the number as a primary quality (1690, Book II, Ch. viii). Indeed, the bed in my room has the property of being one, independently of the sensory system of the perceiver. For applied small natural numbers, this seems to work: I can perceive two, three, or even six coins at a glance; and some savants can perceive more than a hundred coins at a glance, although they are also unable to perceive large numbers without counting. Penelope Maddy noted that the ten fingers of her hands are located on her hands (1884: 87). Moreover, applied numbers can even move together with their bearers. For instance, the 26 stones that make up Stonehenge were originally transported to Wiltshire from Wales.

   Although these applied numbers seem to satisfy our definition of tropes, since they are spatiotemporally localizable, they are surely not at the same level of the counted bed, coins, fingers and stones. Since Frege,we have knownthat applied numbers are dependent on the conceptual ways we have to divide up the world. Thus, I count my 10 fingers but only my 2 hands, we count 26 stones but only a singleStonehenge in Wiltshire. Even if we agree to consider these applied numbers as tropical entities, we must consider them as dependent onthe kind of thing counted, namely, as higher-order properties of concepts associated with the higher-order property of existence (Frege 1884: sec. 55)… Now, understanding the concept (differently from Frege) as a conceptual rule, not as rule actualized in someone’s mind, but as a dispositionally localizable trope, we can read existence as the (higher-order) property of this dispositional rule of its effective or warranted application. Moreover, this effectively possible application of a rule must be also a higher-order localizable dispositional trope-property indirectly belonging to what is said to exist, since it will be the higher-order tropical property of such a dispositional conceptual rule of being applied in some domain, which the real application of this conceptual rule proves. (Because of this, a thing can exist even if its conceptual rule does not actualize, insofar as if this rule were actual it would be effectively applicable to it, which makes the existence of things independent of the actual existence of cognitive subjects or rules.)

  Now, when we count different applications of a conceptual rule, what we get are applied numbers added to existence.Consider, as Frege would do, the case of the empty concept of ‘moons of Venus’. Symbolizing the conceptual rule for ‘moon of Venus’ as V, we can symbolize the idea that Venus has 0 moons as ~∃x (Vx). The concept ‘moons of Earth’ applies to only one object. Symbolizing the conceptual rule expressed by ‘…moons of Earth’ as E, we can render the idea that the earth has onemoon as ∃x [Ex & (y) (Ey → y = x)]. Symbolizing the conceptual expression ‘…moons of Mars’ as M, we can symbolize the idea that Mars has twomoons as ∃x [(Mx) (My) & ~(x = y) & (z) (Mz → (z = x) v (z = y))]… Important is that these existences and applied numbers can be seen as spatiotemporally localizable higher-order possible properties or tropes belonging to possible first-order tropical possible conceptual rules. The number 2 associated with the concept ‘moons of Mars’ applies to Phobos and Deimos, what means that this applied number is located presently and in the surroundings of Mars and not in a distant Galaxy in a remote future or in a Platonic heaven. Our only moon exists circling the earth and this existence is not in Andromeda. Finally, there are the 0 moons of Venus,in the sense that the conceptual rule of ‘moon of Venus’ isn’t effectively applicable to its surroundings as expected.

  After identifying the applied natural number with the dispositional tropical property of the enumeration as an idealized counting of the effective application (existence) of dispositional tropical conceptual rules, what we need to do is to separate (or abstract) the numerical trope from the dispositional tropical conceptual rules in question and the dispositional application of them as attributions of existence. I suggest wecan represent in a separated form these numerical tropes as spatiotemporally located dispositional sets-tropes of applications that in this way represent applied numbers separated from their associated conceptual rules. So, we can represent the numerical property of not being locally applied of the conceptual rule of ‘Venus’moons’ using a set containing as its only member the lack of the applicability of an attribution of existence or {~a} (= 0). We can represent the numerical property of being locally applied only once,using the set that contains only one application of some conceptual rule or: {a} (= 1). We can represent the property of being applied twice using the set that contains a double trope of application or {a, {a}} (= 2). We can represent the property of being applied three times as the set {a, {a}, {{a}}} (= 3), and so on. Since {a, {a}, {{a}}} (=3) contains {a, {a}} (=2) that contains {a} (=1), each new representation shows the right kind of complexity,containing its antecedents. Moreover, it is essential to see that althoughwe are here distinguishing applied numbers by means of sets, these are not pure sets! (We are not trying to build natural numbers from nothing, as do mathematician like von Neumann or Zermelo, beginning with the null set.) We are dealing with spatiotemporally located sets of higher-order trope-properties of ideally countable applications of conceptual-rules (I say ideally countable because great natural numbers are not countable to human beings). The two members set-trope of Mars’moons is spatiotemporally located around Mars, the one member set-trope of earth moons is spatiotemporally located around the earth, and the null member set of Venus’moons is a set of non-applicability, since there isn’t any moon spatiotemporally located in the surroundings of Venus, as expected – no numerical tropes are to be found.

III

At this point, a mathematician could object that we have explained only applied, not abstract numbers. Mathematicians are not interested in counting the number of fingers ontheir hands. When they perform the addition 7 + 5 = 12, they are considering numbers in abstraction from conceptual divisions of the world. What really matters, they say, are the numbers of abstract arithmetic, the universals, the numbers-in-themselves, independently of their satisfaction by ideally countable material objects. Reducing numbers to numerical sets of located applications, we explain an applied number but not what is common to all equinumerous applied numbers. The natural number 3,formulated as {a, {a}, {{a}}} is a triad, but not what is common to all triads, namely, the universal three, the three-in-itself.

   The proper way to represent what is common to all triads seems to be the appeal to things like a Russellian set of all sets of the same kind. However, this seems to lead us to his conclusion that the universe has an infinite number of objects (1919, Ch. XIII), whichis too demanding,considering how infinite an actual infinite must be. Moreover, without assuming tropes,these sets seem to be abstract objects. In any chosen way, we seemto overpopulate our world with an infinite number of abstract objects.

  However, in the same way as we have constructed universal quality-tropes without appealing to sets, we can also construct universal numerical tropes without appealing to sets. As we have seen, an applied natural number can be understood as a trope, since it is spatiotemporally localizable as a second-order numerical property of a dispositional conceptual rule resulting from its (ideally countable) applications. Consequently, in order to account for the universal as the number-in-itself, we can also appeal to our disjunctive device.

  In this case, for instance, it is conceivable that the number 2 in itself would be a disjunction between a set instantiated by a chosen located higher-order set-trope of enumerable dispositional applications used as a model (e.g., the number 2 in the statement ‘I have 2 hands’) or any other strictly similar (equinumerous) located set-trope. Now, in order to get the number 2 as the ‘abstract universal’, the ‘2-in-itself’, all we need to do is to apply the same procedure we have applied to get universals from our usual quality-tropes to a numerical set-trope of located applications {a, {a}} understood as a selected model. For instance:

Number 2 (Df.) = a selected model understood as a higher-order set-trope of located countable applications {a, {a}}*, or… any further higher-order set-trope of located countable dispositional applications strictly similar (equinumerous) to {a, {a}}*.

In this sense, the natural number as a universal (or ‘abstract object’) can be defined as:

A higher-order located (ideally) countable set-trope taken as a model or of any higher-order located (ideally) countable set-trope strictly similar (equinumerous) to the model.

Note that such a universal or abstract object remains empirical since it is a higher-order disjunctive numerical set-trope of located dispositional applications that can be found as something dispersed across our whole spatiotemporal world.

  Assuming a definition like that, we neither stumble over controversial pure or infinite sets nor seem to remain limited to particular instances or directly committed to any empirical constituent of material objects. The foreseeable conclusion is that even the abstract world of mathematics is built of thin higher-order tropes. Such tropes, like some others, would be situated at the peak of a building whose originating genetic-epistemic foundations are our more feasible perceptually given quality-tropes, although retaining the distinctive property of being spatiotemporally located, even if in an extremely diffuse way. A final advantage of the proposed view is that it is consonant with the applicability of mathematics to the empirical world. Mathematical entities are applicable because, like our world, they are made up of tropes.

 

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