TROPES AND UNIVERSALS (DRAFT)

 DRAFT

 

UNIVERSALIZING TROPES: A SIMPLE WAY

 

 

Summary:

Donald Williams suggested that under the assumption of a trope ontology, universals can be understood as sums or sets of tropes. My goal in this article is to show that there is an easier and much more convincing way to deal with the issue. It consists of returning to the root of the problem, namely, the predicative problem of how to say the same of many. I intend to show that this is the real problem and that trope theory does not demand the resource of universals to solve this problem. All we need is to develop a universalizing ability, namely, the ability to identify precise similarities between tropes used as models and experientially given tropes.

Key Words: trope theory, universals, ontology, Donald Williams.

 

My goal with this article is to present a solution to the problem of universals using trope theory. I claim this solution is easier and more effective than Donald Williams's troublesome solution, fostering the trope theory considerably. In the first section, I begin by exposing the basics of trope ontology as motivated by Williams. In section two, I present his solution to the problem of universals by means of tropes and its difficulties. In section three, I present the original Socratic-Platonic problem that leads to the problem of universals. In section four, I present my own solution to the original problem by means of tropes, which by-passes the problem of universals as a pseudo-problem. In section five, I show how my solution applies to more complex tropes, and in section six, I briefly conclude.

 

1. Which kind of things are tropes?

The theory of tropes resulted from an idea as brilliant as simple by the American metaphysician Donald Williams, developed in the classic article "The Alphabet of Being" (1953 I-II). His fundamental metaphysical thesis was that:

 

Any possible world, and hence, of course, this one, is completely constituted by its tropes, and their connections of location and similarity and any others that may be (1953 I: 8)

 

In other words, tropes are, for Williams, the universe-building pebbles. But what are tropes? He defined them as abstract particulars because we can abstract them from concrete particulars. For example, he used the red of a lollipop, its shape, its sweetness, its smoothness, its hardness, its weight... Although this definition is technically correct, opposing tropes as abstract particulars to concrete particulars such as the lollipop, a more useful way to say the same is to define tropes as any spatiotemporally localizable properties. This contrasts with the use of the concept of property in the philosophical tradition, in which it is generally understood as an abstract entity in the sense of not being spatiotemporally localizable.

   Williams’ understanding of the extent of the concept of trope is made more evident by the following list that I take from his article:

 

Color, shape, surface, odor, red, size, triangularity... pain, love, sadness, pleasure, emotion, belief, serenity, perception, discrimination, intention, disposition, power, mental processes, sequences of thoughts... a smile, a sneeze... An election, a musical performance, a love affair, a moral decision, an act of contrition, a piece of impudence... the beauty of Mary, Mary being beautiful, the figure of a woman, her complexion, her digestion...

 

This list shows us that tropes can be simple properties (that red, a twinge of pain, the whistle of a train) or compound (a mental process, a sneeze, a love affair). They can be external (a sneeze, a smile, the figure of a woman), internal (sadness, serenity, pleasures, dispositions, thoughts), or even mixed (an act of contrition, a piece of impudence). They can also be homogeneous (a violin solo) or heterogeneous (a piece of impudence, Mary’s digestion).

   Sticking to the definition of tropes as spatiotemporally localizable properties, it is pretty clear that also very vague dependent entities, such as the forces of nature, can be considered tropes since they are localizable properties, albeit in a more or less dispersed way. Thus, electromagnetic forces, strong and weak forces, and a gravitational field are all tropes since all these are spatio-temporally localizable properties that can be abstracted respectively from atoms in motion, from the interaction between quarks and gluons in the atomic nucleus, from the interaction between subatomic particles in atoms, and finally (in the case of gravity) from the bending of space-time in the vicinity of massive bodies.[1]

   For Williams, tropes only contrast with concrete particulars such as, to use his examples, Mary, a church, and a nation. He analyzed concrete particulars as sets of concurrent or compresent tropes in the sense that their members are co-localized and co-temporal. They can be internally organized, like a specific chair or a particular human being, or quite disorganized, as in the case of a rock or an asteroid. Although concrete particulars are usually medium-sized dry objects, they can be as small as protons and atoms and as large as stars and galaxies. (Although Williams does not consider it, it is worth adding here that the criterion for identifying concrete particulars differs from the criterion for identifying tropes in that normally only tropes can be predicatively designated, in addition to their typical existence in dependence of concrete particulars – these are Aristotelian insights that, despite their allusive relativity, are worth to be preserved.)

 

2. Williams’ solution to the problem of universals

In addition to trying to solve the question of concrete particulars by resorting to tropes, Williams wanted to do the same with the question of universals. Here's how he introduced it:

 

Speaking again approximately, the set or sum of tropes precisely similar to a trope, say, that red, can of course be, or at least formally correspond to, the abstract universal or 'essence' which can be said to exemplify a definite sample of red. (1953 I: 9)

 

This somewhat far-fetched definition is usually presented more clearly: a universal (essence) is a set or sum of tropes that are precisely similar (i.e., qualitatively identical). The universal of the color red, for example, is nothing more than the set or sum of tropes precisely similar to a given trope of red. (Cf. Campbell 1997: 484). When I say that red is a color, what I mean is that the set of all the tropes of red (the universal-essence of red) is contained in the set of the tropes of color (the universal-essence of color).

   There are problems with Williams' solution. The most discussed has been that of the regress to infinity of tropes of precise similarity (see Campbell 1990: 34-37). If the world is made up of its tropes, then it seems that precise similarity is also a trope. In this case, the precise similarities between the tropes will need to be precisely similar, requiring a new class of precise similarities. However, the tropes of this new class of precise similarities will need to be precisely similar, requiring a third class of precise similarities, and so on, so that the result seems to be na infinity regress.

   There are several ways to address this objection. Campbell, for example, suggested that identity is an internal relationship between tropes that can only count as a pseudo-addition with no real ontological basis (1990:37). Personally, I prefer not to harbor any prejudice against the thiness of any trope. I like to think that precise similarity because it is a relation dependent on the existence of these tropes, is itself a trope insofar as it is findable between them and through them and not something that is found on the other side of the universe or anywhere (that nothing exists anywhere is an axiom of the theory of tropes). And the same, I would say, of a precise identity between two precise identities. If the latter two are somehow locatable, then the precise similarity trope between these precise similarities is also somehow locatable. The regress, therefore, exists, but it is not vicious because nothing forces us to go forward (Cf. Maurin 2007). It is like the regress we find when we say, "If P is true, then it is true that it is true that P is true." We can go on interspersing 'it is true that', but nothing prevents us from stopping at 'P is true', since continuing the regression adds nothing to what we want to say. Similarly, we can stop by saying that a class of precisely similar tropes has a class of tropes of precise similarity between its tropes, but that this does not oblige us to appeal to a subsequent class of precise similarities between these precise similarities.

 Nevertheless, my difficulties with Williams's solution are of a different order. Not only do we not know whether the set of precisely similar tropes is itself a trope, but sets have sizes, and can increase or decrease in size, while universals do not appear to have size, especially if they are essences, as he intended. The biggest problem, however, which in my judgment is decisive, is that sets of tropes that are precisely similar to each other are usually too large to fit in our heads. In other words, in the vast majority of cases, the set of tropes is cognitively inaccessible to us. Consider, for example, the set of precisely similar tropes of red: we can say that it exists, but not that anyone has ever met it. And if it is an essence, then that essence is unknowable. Someone once suggested to me that they are open sets. But open sets only seem to exist in the mind and not in nature. Also, how many members does this open set have? Can it be a unitary set?

 

3. The problem of how we can say the same of many

The problem of universals arises from another problem, which is how we can say the same of many, which could be called here the problem of predication. It is a question of knowing how it is possible for us to apply the same general term to a multiplicity of things, sometimes very different. How is it possible, for example, that anyone can say that Socrates was just, that Nicholas Winton's noble action was just, that the Finnish political system is fairer? The question is linguistic, but it is also epistemological, since it concerns the realization of synthesis in judgments, the cognitive identification of unity in multiplicity.

   The first philosophers to come across this question were Socrates and Plato. Socrates wondered about the definition of concepts expressed by general terms of philosophical interest. Plato famously replied to him with his doctrine of ideas. As he wrote:

 

We are in the habit, I take it, of positing a single ideia (idéa) or form in the case of the various multiplicities to which we give the same name (Republic 596a 6-8).

 

The idea or form is unique and abstract in the sense that it is not spatiotemporal, immutable (eternal), and capable of definition; it belongs to an intelligible world that transcends the visible world by being ontologically independent of it. Plato's suggestion was that by reference to the single idea or form we are able to identify the various things in the visible world that contain imperfect copies of that idea or form (by mimesis), or that in some way share their being with that idea (by methéxis).

   What is important to note is that through his answer to the question of how to say the same of many, Plato moved from an epistemological-linguistic problem to an ontological-metaphysical solution. It was as a result of this movement that the problem of universals arose. Since the idea or form is the universal (to catholou), the problem of universals arises here, which was defined by Aristotle as that of knowing "that which by its nature is predicated of a number of things" (1984: 17a 38), that is, the problem of the ontological nature of something objective that by itself makes us capable of saying the same about many. For realist philosophers such as Plato and Aristotle, under the more usual interpretation of the latter (see Metaphysics 1034a 5-8) universals are abstract, immaterial entities, even if, in Aristotle's case, they are generally dependent on the existence of the things that constitute the visible world.

   But it was not only the realist philosophers who moved from the epistemological-linguistic problem to an ontological-metaphysical solution. There are also nominalist philosophers who have followed this same path. This is the case of class nominalists, for whom the general term has as its reference the extension of the things (concrete particulars) to which it applies (Armstrong 1978 ch. 4; 1989, ch. 1). A class nominalist might say that the class of all red objects must be responsible for the applicability of a general term like 'red'. And a sophisticated class nominalist like D. K. Lewis (1986: 50 ff.) would answer the problem of different general terms with the same denotation and different connotation by appealing to their application to the class of objects that encompasses all possible worlds, since in that case these classes would no longer need to coincide.

   It seems to me quite likely that in proposing his solution to the problem of universals as being classes of tropes precisely similar to each other, Williams was being inspired by class nominalism, an idea that had already been floated well before the publication of his paper. He only replaced things, concrete particulars, with tropes as members of the class considered to be the universal.

   I call the kind of solution of the problem of how we can say the same of many by recourse to universals such as those considered above the "traditional solution." My opinion is that it was nothing more than a great error that compromised more than two thousand years of metaphysical research without substantial results, namely: the error of offering an unnecessary ontological-metaphysical solution to a linguistic-epistemic problem. Although this mistake was first made by Plato, it gave rise to a tradition of investigating universals so ingrained that its epistemic origin was forgotten and it became very difficult to get it out of the heads of philosophers.

   Not all philosophers, it is true, have fallen into this trap. Through their particularism, Berkeley (1973 Intr.) and Hume (1978 I, I, VII) were honorable exceptions. For Berkeley's immaterialism, as is well known, the world (besides that of spirits) is totally constituted of particular ideas of mental natures, having nothing more to do with Platonic ideas or Aristotelian universals. The things of the so-called external world are nothing more than more intense and organized ideas that impose themselves on us (1973 I, sec. 30). An idea can be general, in the sense that it applies to many and diverse things. But for him they are never abstract, which means that they never have the function of Platonic or Aristotelian universals or anything else other than their purely mental and in this case also subjective nature. As he wrote:

 

It seems that a word becomes general by being made the sign, not of a general abstract idea, but of several particular ideas, any one of them indifferently suggested to the mind (1973: Intr. Sec. 11)

 

That is: a general term is associated with mental ideas, for example, the general term triangle is associated with images of several particular triangles, equiangles, rectangles, obtuse... On this basis we are able, whenever we have access to a given triangle in the external world, that is, to an external idea of a triangle, to compare it with an idea of the ideas that we have associated with the general term and apply this term to the given triangle in the so-called external world. To explain this operation – which is nothing more than that of identifying it in many – Berkeley does not need at any time to resort to universals. He only compares the idea from which he has the name with the idea given to the experience, and when he perceives the similarity between them, he gives the same name to the idea given to the experience. Hume hailed this proposal by Berkeley as a major discovery made in the republic of letters (1978: 17). I can anticipate that my strategy will be largely structurally analogous to Berkeley's, for what I will do is essentially replace "idea" with "tropes." However, I will be free from the main criticism of Berkeley's solution to the problem of predication, which is that it is an imagistic solution and that many concepts do not require the formation of images or secondarily require them. Since tropes need not be essentially imagistic, even if they must be somehow derived from sensory experiences, we will not be open to this accusation.

   In what follows, I want to propose my solution to the problem of how we can say the same of many, inspired by the Berkeley gambit. Although it is based on the theory of tropes, this solution does not need to answer the epistemological problem through an ontological solution. It is an alternative solution that dispenses with the appeal to universals in the sense considered. In other words, my proposal will be purely epistemic: universals are an unnecessary ballast that has only served to confuse philosophers in the last two millennia. It's time to get rid of them! It is time to say goodbye to Platonic-Aristotelian entities common to many individuals, classes of objects or their extension into possible worlds, or even to sets of tropes that have taken on a life of their own by the chance fact that they have been used in an inevitably mistaken explanation of how we are able to say the same of many.

 

4. Universalizing Rule Following Skills

I propose that we learn to say the same about many by gaining a skill. It is the ability to (i) associate certain tropes with the corresponding general terms and (ii) use those same tropes (or any others that are precisely similar to them) as models, to compare with any trope that we are given to consider, to be able to tell whether or not it has a precise similarity with the model.

   Once this skill is acquired, we become able to say the same of many without resorting to universals or classes of tropes as universals. Out of reverence for the immense efforts that philosophers have made to solve the problem of universals, I will call it the universalizing ability.

  I want to start with a very simple example. Suppose Mary has taken a painting course and learned to identify the color 'Terra de Siena'. To this end, she had eye contact with many examples of this color, having learned to associate these examples with the name of the color and becoming able to remember precisely this model or trope of color. Later she takes a trip to Italy. There she identifies a number of buildings painted with the same color as Terra de Siena. All he needs to do to identify the color of these buildings correctly is to have the memory of a certain trope and to be able to identify new tropes precisely similar to it. She developed the ability to recognize the Land of Siena trope. She learned to say the same of many. And obviously (to her own luck) she does not need to resort to any immense set of Land of Siena tropes that are precisely similar to each other to be able to use this general term predicatively. Maria's universalizing ability with respect to the Terra de Siena trope can be specified as that, having associated with models of tropes of the same color that she holds in her memory, she has become able to identify any Terra de Siena trope that is given to her experience as being precisely similar to those model tropes. We also say that Mary learned to use the word 'Land of Siena', that she showed that she knew its meaning as a rule of use, in other words, that she mastered the concept or conceptual rule.

   Two points are worth mentioning. The first is that memory is something secondary to the universalizing ability. Not that memory is a problem. It has nothing mysterious about it. It is like a copy of something, a copy that we learn to trust because it works well enough to be considered trustworthy, since we are able to confer that trustworthiness by interpersonal comparison between what has been identified and what has been reidentified. But it is important to highlight that it is an intermediary that is not part of the procedure as such. The second point is that this ability is always, at least in principle, capable of generating an extensional class of tropes—but a class that has no relevance whatsoever in solving the problem of predication.

   To highlight these points, imagine that a group of people with a certain mnemic deficiency are in an exhibition and that they are given a task: to identify the paintings in which the color Terra de Siena appears. There are 12 numbered frames. Each of them receives a table where the name 'Land of Siena' is found next to a circle painted in the same color and twelve squares corresponding to each of the 12 paintings. All they need is to mark the spaces corresponding to the frames in which they find the color Terra de Siena. By taking the tables with their painted circles close to each frame they will be able to mark the spaces corresponding to the frames in which a color precisely similar to the model is found. By doing this, people are carrying out a universalizing practice without resorting to memorized models. Finally, let's say that the only frames with these color tropes are 2, 3, 8, and 12, and that this shows up as a result in the correctly populated tables. With this they found, within a very narrow scope, a set of Terra de Siena tropes that are precisely similar to each other: the extensional set {2, 3, 8, 12}. However, this set has no explanatory force with respect to the ability to say the same of many, here mechanically practiced even without appeal to memory.

   One characteristic of what the people in the example do is that they are following a rule that tells them to compare the color model trope of the table with the color tropes found in the frames in search of a precise similarity. The model trope that they see and that we are able to actualize in our memory is what serves as a criterion for the application of the rule. The rule is applied when people mark the square corresponding to the trope that is on the board on which the Terra de Siena trope is found. Since we say that this person now has the concept of the Land of Siena, there is nothing more natural than to consider this universalizing ability a conceptual rule or concept.

   Returning to our problem, we can now define the universalizing ability as follows:

 

Universalizing ability with respect to a general term (Df.-U) = ability to, having associated the term with a model trope any Tm, be able to use it as a criterial rule capable of allowing the identification of any given trope as being or not being precisely similar to Tm.

 

The universalizing ability is a conceptual rule that corresponds to what we ordinarily call a concept. And this rule, like everything else, is a findable trope in every application of yours and nothing that is metaphysically abstract.

 

5. Universalization of complex tropes

 So far we have explained our universalizing ability relative to a trope that can be classified as perceptually simple, which is the color Earth of Siena. But what about more complex cases, such as the other tropes cited by Williams, tropes such as those of triangularities, of sneezing, of smiles, of sadness, of a thought, of an act of contrition?

   I believe that at least for those who have considered the constructive procedures of a philosopher like John Locke (1979: book II), demonstrating that we can start with simpler ideas and combine or separate them (by abstraction) in order to form more and more complex and diverse ideas, this would not be seen as a real problem. The difference is that here we do not start with ideas, as in Locke, but with something more general (which includes what he called qualities), that is, with simpler tropes, and then proceed to combine them in order to form more complex and diverse tropes, to such an extent that they may appear to recur to universals independent of empirical foundation. We use the memory of these tropes as a criterion for identifying precisely similar tropes that are in some way given to us by experience. And in doing so we are again applying the same universalizing skill exemplified above, which is the ability to follow conceptual rules or, more simply, to apply concepts.

   I want to analyze some of these tropes in an attempt to demonstrate that they are spatio-temporally localizable particulars and that they have nothing abstract in the realistic (Platonic-Aristotelian) sense. To do so it will be necessary to define or decompose them analytically in order to demonstrate that although they are not as directly and immediately identifiable as the earth trope of burnt Siena, their origin results from simpler trope combinations found in sensible experience than is spatio-temporally given. The goal is to make it plausible that the universalizing ability acquired in the cases of more complex tropes considered by Williams does not differ appreciably from the newly defined universalizing ability.

   Let's start with the easiest case of a complex external trope of sneezing. It can be taken for granted that it is on the basis of interpersonal corrections of positive and negative examples that we usually learn tacitly the concepts expressed by the most common general terms of ordinary language (Tugendhat 1976, lesson 11). Thus, in general, we learn what sneezing is by observing adults give this name to the act of other people or ourselves of sneezing, being praised when we get it right and corrected when we make a mistake when applying the word. In this way, we acquire the universalizing ability that allows us to identify sneezes as precisely similar to models already experienced. This is also about learning the concept (the rule of conceptual application) based on our memories of sneeze tropes. As this rule is tacitly learned, we do not generally know how to verbalize it. But dictionaries help us, defining sneezing as "an involuntary movement of the airways to expel any foreign substance from the body." This definition makes explicit what the irrelevant sneeze trope is.

   Let us now look at the relatively simple trope of triangularity, also recalled by Williams. Here learning is also done through interpersonal examples in a tacit way. We are presented with equiangular triangles, acute, obtuse, rectangles, until we learn, not an abstract triangle common to all of them (as Berkeley would have done), but to vary the internal angles of a model triangle so that we are able to form the corresponding images of triangles given in the experiment. Only later do we learn definitions such as that of "a three-sided polygon", which decomposes the internal tropes of the triangle into that of a polygon, which is also decomposable.

   Now consider the internal trope of sadness: all we need to universalize is to be able to identify instances of sadness as being precisely similar to tropes of sadness that we have already learned to identify in other people and in ourselves and that we use as a model, that is, as criteria for the conceptual rule of identifying sadness. Dictionaries characterize this trope as a feeling of despondency, melancholy, unhappiness, and hopelessness.

   Let us now look at the mixed and rather complex trope of the act of contrition. I have been informed that it is a Christian prayer that asks for repentance for sins and mercy from God. All we need to know to know how to universalize is to have some example of an act of contrition in memory or, in my case, to be explicitly informed about this trope so that I can imagine it in such a way as to be able to identify cases precisely similar to it.

   I want to consider now two examples of countable name tropes that were not considered by Williams. They differ from the previous ones in that they concern the "essence" of concrete particulars. A first example is that of an artifact name: the trope of chair (from "to be a chair" or "chair"). This is a complex external trope. I think I can give a reasonable definition of a subject (which applies to both Portuguese and English):

 

Chair trope (Df.): non-vehicular seat trope with backrest, made for one person to sit at a time.

 

In fact: if it had no backrest, it would not be a chair, but a simple bench. It cannot be found in vehicles such as cars, planes, trains, because in that case they will be called seats. They need to be made for one person to sit on, otherwise they would be sofas. Even if a chair were teleported to a planet where people were extremely thin and several of them could sit in the same chair, it would not turn into a sofa, as it is an artifact made for one person to sit on at a time. And a chair that has not been made as an artifact, say, a chair carved into the rock, is a chair only metaphorically, no less than a toy chair. And armchairs are specimens of chairs. All this we learn through perceptual experiences. Note that we are not able to have the direct perception of a chair trope. It is not written on a chair that it is made for one person to sit at a time, or that it should be a bench with a backrest. Nor that it may have been the work of a carpenter.

   That is why it took a good number of experiences of positive and negative examples so that we could gradually tacitly learn the complex trope, linguistically expressed in the definition of chair. Therefore, keeping in mind the external, complex and articulated trope that we associate with the word chair, we are able to identify precisely similar tropes of "chair", such as those that exist when we encounter table chairs, wheelchairs, beach chairs, electric chairs, thrones... Of course, this trope need not be reduced to something of which we can have a mental image. Neither tropes nor the procedure of universalization force us to a purely imagetic conception of reality.

   Someone may now ask a challenging question: how to distinguish the trope of chair – which I will call the trope of "chair" – from the concrete particular that is a certain chair? After all, the chair trope is a set of spatio-temporally locatable tropes present, and where a chair is found, one will also find a complex trope of its chair. The answer lies in the fact that a chair, as a concrete particular, is constituted by many other tropes, in addition to those that constitute its chair. These are accidental tropes that do not define it, but that help us identify the concrete particular. For example, the chair I'm sitting on now has a soft upholstery, it's made of plastic and wood, it's a faded yellow color, it's old, it was bought in a second-hand furniture store... These are not elements of the trope of the chair in which I sit, but of this concrete particular, in addition to being what it is, it instantiates this trope. Hence, the concrete particular is a set of present tropes that extrapolates the trope that specifies it. This answer can also be applied to other countable names of "essences" like the one in the next example.

   Let us consider, finally, a much-talked about trope of the countable concept of natural species: that of the human being, traditionally defined by the Greeks as that of a rational animal, but characterized today in dictionaries as a bipedal primate endowed with reason, culture, and advanced language. Of course, this characterization of the complex trope cannot be directly read in the immediate sensible perception we have of other human beings, as with a color trope. But it is also learned tacitly through positive and negative examples, even if we are not able to linguistically articulate the dictionary definition. I myself have seen many of these bipedal primates during my existence and have been led to recognize myself in the mirror as one of them. I know that they are capable of a certain degree of reason, culture, and advanced language. It was through this public learning arising from extraordinarily complex articulations of simple perceptual tropes that I gradually learned what human beings are and that I became able to use the complex trope associated with the general term 'human being' as a rule in recognizing precisely similar trope articulations in the cases of the most diverse beings recognizable as humans. I do not need or am able to form images of this trope, since it is not imagetic. But he induces me to form associated images, like that of an Oriental man. Influenced by Locke's analyses, we would say that it results from an extraordinarily complex articulation of tropes whose ultimate elements are perceptually experiential and imagetic in a broad sense of the word.

 

6. Conclusion

The proposed solution seems philosophically plausible if we restrict ourselves to the examples of tropes presented and discussed in this article. It is entirely in line with Williams’ metaphysical project of using the concept of the trope as an epistemic basis for the construction of reality. It excludes the paralyzing difficulty that I pointed out in his solution of the problem of universals and makes the ontology of tropes more palatable.

   Indeed, most of the questions are still open. There are properties whose universalizing ability does not seem to be prima facie covered by examples such as those discussed here. Williams did not question whether spatial relationships can count as tropes. He also left open the case of mathematical properties such as classes and numbers or the property of existence... which seem abstract in the sense of not being perceptually localizable...

   I believe that much of this could, in principle, be explained in terms of spatiotemporally localizable properties, that is, tropes. Consider, for example, that the only way to justify the application of mathematics to the empirical world seems to be that mathematical entities can be considered somehow to belong to that same spatio-temporal empirical world. As a bet on a radical empiricist naturalism, the ontology of tropes would be, in principle, available. However, any more ambitious approach to these issues is beyond the scope of a brief article.

 

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Williams, D. C. (1953a) “On the Elements of Being I.” Review of Metaphysics, 7(1) 3-18,

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[1] Keith Campbell (1990: 151-155) has called them quasi-tropes, but conceding the definitions given it does not seem that these forces need be less than tropes.