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Universals

Universals are the fundamental entities in metaphysics.  Their existence is implicitly assumed in language, mathematics, and science.  Yet their exact nature is hotly debated.

Table of Contents

What Are Universals?
All languages, both natural and mathematical, are based on two fundamental types of words, those that describe things (subjects and objects), and predicates.  We have a clear and intuitive understanding of things; we can see, feel, touch, hear, smell, and taste them.  But what about predicates?  What is redness?  When we say that “Haile Gebrselassie is a runner”, what is “runner”?  Those things to which predicates refer are known in philosophy as “univerals”.  There is a great deal of controversy regarding exactly what universals are, where they exist, or if they even do exist1.

Universals are so fundamental to our thinking that we take them for granted without giving them a second thought.  We all speak of “red”, the number 4, and iTunes as if they exist, and as though they are single entities with unique identities.  Indeed, all of science is based on mathematics, and mathematics is the study of nothing but universals.  For this reason there is a significant amount of controversy concerning the ontological nature of mathematical objects2.  If fractals exists, where are they?

Perhaps the most famous theory of universals was Plato’s forms3.  To Plato, universals are perfect representations of what is found in the real world, but that existed in some other, non-physical realm.  These forms are manifested in the real world as imperfect copies of their true selves.  Thus, the Olympic champion Haile Gebrselassie is an imperfect copy of the perfect form of a runner in the realm of forms.  Many mathematicians are Platonists.  Aristotle also had a theory of forms, but to him, forms were real things in the real world, used to shape “matter”, or the stuff of which objects are made4.  For Aristotle, everything has matter and form, with the former corresponding roughly to subjects and objects, and the latter to predicates.

Patterns
In my view, Aristotle’s matter and form, in modern terminology, constitute physical objects and the relations between them.  The most basic physical objects would be the fundamental, elementary particles such as quarks and photons5.  Everything else that we experience would be constituted by the relations between these particles.  Thus, elementary particles such as quarks could be arranged in ways to create larger particles, such as protons and neutrons.  These composite particles in turn are the matter for further relations, such as those that constitute atoms.  In this way, the world we experience is constructed from elementary particles by consecutive layers of relations.

This view explains the philosophical dilemma of how such a rich and varied world as the one in which we live can be built from monolithic particles that physics tells us are indistinguishable from one another.  The answer is that everything that is interesting in the world is due to the varied relations between these particles.  Another way to think about these relations is as information or patterns.  The universe is not interesting because of its matter, but because of the information about objects, or equivalently, the patterns that exist in the universe.

I will refer to relations and attributes (which apply to only a single object, as opposed to relations, which apply to two or more objects) using the somewhat unorthodox term “patterns”.  Patterns are closely related to the mathematical theory of sets6.  Patterns, such as “red” or “bigger than”, are shared by multiple individual objects.  For every pattern, there is a set, or collection of things, that corresponds it.  The set corresponding to a particular attribute is the set of all objects that has that attribute.  Likewise, the set of all objects that corresponds to a relation is the set of all ordered sets, or “n-tuples” to use the correct jargon, that belong to this relation.  Thus, since San Diego is north of Tijuana, the ordered pair “San Diego, Tijuana”, or <San Diego, Tijuana> to use set theoretical notation, is a member of the set “north”7.

There is a fundamental difference between sets and patterns, however.   Sets are said to be the same if they contain the same members, whereas patterns are not.  For example, the set of all birds and the set of all things that has feathers is the same, whereas the attributes “is a bird” and “has feathers” are different.  In mathematical and philosophical jargon, sets are said to be extensional, whereas patterns are set to be intensional.  Indeed, all set theories contain the Axiom of Extensionality, which states explicitly that sets that contain the same objects are equal.  While extensionality is useful for mathematics, and should probably be true for attributes and relations that are necessarily the same, such as “2+2” and “4”, it is philosophically problematic for other attributes and relations.  The attributes “is a bird” and “has feathers” are clearly different things, with the latter being a feature of the former.  The sets of things that have these two attributes may be the same today, that does not mean that this will always be true in the future.

So where are universals?  My opinion is that, just as things maintain their identity at different points in time, so different instances of a particular pattern constitute a single identity.  Thus, iTunes is a universal that is the pattern, the information, that is embodied in every copy of iTunes.  That pattern maintains its identity in every physical instance in which it is manifested; i.e. on every computer on which it is installed.  The situation is slightly different for sets, in which the universal is literally the collection of all things that belong to the set.  To me, the things that have a particular pattern are superfluous; it is the pattern that constitutes the attribute or relation that is important, that gives the universal its identity.  I believe that this concept of patterns as universals, that apply to individual objects but are not embodied by them, is a rigorous, modern version of what Aristotle had in mind with his concept of form and matter.

Problems with Patterns
Unfortunately, the theory of patterns, and indeed set theory, has numerous philosophical problems:

  • As mentioned above, the Axiom of Extensionality is problematic.  For philosophical reasons, it does not appear to belong in a theory of universals.  However, its absence would disallow some theorems of mathematics8.
  • It is well known that naive set theory, the theory based on the assumption that for any formula, a set exists of all objects for which the formula is true, is inconsistent9.  Many set theories have been devised that can be shown to be consistent, but there is no clear way to choose between them10.  Furthermore, some sets that exist in some of these theories do not exist in others.  For example, the definition of the number 2 will vary from theory to theory.
  • Not every pattern or set that can be defined exists.  In other words, there are definitions of sets that cannot possibly exist, because their very existence would be a contradiction.  This is essentially the same as saying the naive set theory is inconsistent.
  • Not every pattern or set that exists can be defined.  George Cantor demonstrated at the end of the 19th century that there are different sizes of infinity11.  It can be proven that the number of logical formulas, and therefore the number of possible definitions of sets, is only “countably” infinite, meaning that their number is the lowest of all levels of infinity.  On the other hand, the number of possible sets is an infinity so large that it cannot easily be characterized, because for any level of infinity N, there is a larger infinity equal to 2N, and this series seems to increase without bound.  The conclusion is that the vast majority of sets or patterns actually cannot be defined.
  • Any rigorous theory of sets or patterns is necessarily incomplete.  This was demonstrated by Kurt Gödel’s famous Incompleteness Theorem published in 193112.  What this means is that there are some sets or patterns that exist, but whose existence we cannot prove from any finite set of axioms.
  • The concept of ordering is problematic.  As mentioned above, orderings are necessary to describe relations, or any set or pattern that involves more than one object.  Furthermore, there are numerous possible ways to define orderings, but it is common in mathematics to adopt an arbitrary definition of an ordering7.  I believe that this is not a problem, however, in a theory of universals, as the place of the objects in a given pattern ought to be a primary feature of the pattern itself13.

Advantages of Patterns
Despite the problems listed above, I believe the patterns are the best rigorous characterization of universals that is available.  The concept of a pattern as having identity seems to be the most plausible way to state what, exactly, universals are.  The concept of universals as patterns is a completely naturalistic theory; it does not require the existence of any supernatural or metaphysical realm.  It is also an intuitive one that most if not all people understand.  The hierarchy of construction of larger objects as patterns of smaller objects helps explain why such a rich and varied universe can be constructed out of a very small number of classes of elementary particles.  Finally, this concept explains why the universe can ultimately be explained using mathematics: with universals as patterns, they are essentially the same as the objects of mathematics.

A valid theory of universals is essential as a foundation for the validity of mathematics, natural language, and our very thoughts.  Universals provide the structure on which all things are based.  While it faces some definite philosophical difficulties, the theory of patterns is by far the best candidate for a workable theory of universals.

End Notes

  1. For a discussion of the various theories of universals that have been proposed, see Klima, Gyula, “The Medieval Problem of Universals”, The Stanford Encyclopedia of Philosophy (Fall 2013 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/fall2013/entries/universals-medieval/>.
  2. For a good discussion of the ontological status of mathematical objects, see Körner, Stephan, The Philosophy of Mathematics: An Introductory Essay, Dover Publications, New York, NY, 1960.
  3. “Theory of Forms”, Wikipedia, URL=<http://en.wikipedia.org/wiki/Theory_of_Forms>.
  4. “Hylomorphism”, Wikipedia, URL=<http://en.wikipedia.org/wiki/Hylomorphism>.
  5. Of course, our understanding of the list of the most elementary particles may change if physicists find better theories and even more elementary basic particles.  Some physicists even think that there are no fundamental particles at all; that in the end all that exists are relations, although it is hard to understand how relations could exist with no objects between which to have the relations.  For example, see Kuhlmann, Meinhard, “What is real?”, Scientific American, Vol. 309, No. 2 (August, 2013).
  6. For a good introduction to set theory, see Suppes, Patrick, Axiomatic Set Theory, Dover Books, New York, 1972.
  7. Set theory requires a rigorous definition of an ordered set.  Any definition that could be proven to represent the ordering unambiguously could be used, but the one that is standard for ordered pairs is <x, y> ≡ {{x}, {x, y}}, where {…} constitutes a set.  Orderings with more than two members are built up from smaller orderings; e.g. <x, y, z> ≡ <<x, y>, z>.
  8. A compromise may be to include extensionality only for patterns that are tautologically equivalent.  The philosophical objection to extensionality may not hold for these patterns, and allowing extensionality in this case may preserve most or all of the mathematical results that would be lost otherwise.
  9. This can easily be demonstrated with Russell’s set, the set of all sets that is not a member of itself.  The existence of such a set is impossible, because if it was a member of itself, by definition it would not be, and if it was not a member of itself, by definition it would be.  In symbolic notation, if R is Russell’s set, it can be easily proven from the definition of R that R∈R ⇔ R∉R, which is an obvious contradiction.
  10. For an excellent discussion of leading set theories, see Quine, W.V.O., Set Theory and its Logic, Harvard University Press, Cambridge, MA, 1969 (Revised Edition).  The set theory that is favored by mathematicians is the Zermelo-Frankel theory.  My personal favorite from a philosophical view is Quine’s New Foundations, which is very close to naive set theory, and can be reduced to just three axioms (one being an axiom schema).
  11. See Cantor, George, Contributions to the Founding of the Theory of Transfinite Numbers, Dover Publications, New York.   For a very brief summary of his reasoning, see my essay on the great discoveries of 1879-1931.
  12. Gödel, Kurt, “On formally undecidable propositions of Principia Mathematica and related systems I”, Monatshefte für Mathematik und Physik, 38, 1931, pg. 173-198, English translation available in van Heijenoort, Jean, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA (1967).  For an excellent layman’s description of Gödel’s proof, see Penrose, Roger, The Emperor’s New Mind, Oxford University Press, New York, 1989, pg. 105-108.
  13. This is reflected in the respective notation of set theory and second-order logic.  Consider two objects a and b (in that order) that have a relation R.  In set-theoretical notation this would be written “<a,b>∈R”, whereas in second-order logic this would be written simply as “Rab”.  As can be seen, in the latter case there is no need for the concept of an ordered pair.

November, 2013


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