The Nature of Number

Why Universals are the Ligatures of Reason

Photo by Christian Chen on Unsplash

Introduction

One question that drew me to read philosophy was the question of the reality of numbers. In investigating this question, I discovered that it opens up a whole host of related, and very deep, questions — questions of metaphysics, in fact, that are far beyond my scholarly ability to fully comprehend or elaborate.

But nevertheless I will make a start at why I consider ‘the nature of number’ to be one of the truly fundamental questions of philosophy, and from there, to extend the idea to the nature of universals. I will re-trace my steps from the original insight, and then try to summarise some of the philosophical themes and sources of the discussion I’ve encountered in exploring it. I acknowledge at the outset my amateur status in a field of study which perplexes many experts. Even so, I believe the idea worth exploring, and I hope at least to bring a fresh perspective to it, or perhaps just a way of thinking about it that makes it more approachable for contemporary readers.

A Minor Epiphany

The original impetus for this idea was a minor epiphany. One day it occurred to me, out of the blue, that while the objects of ordinary experience — tables and chairs, pens and paper, people and animals — are composed of parts, and have a beginning and an end in time, the same cannot be said of numbers¹. At the time, I thought ‘Aha! So this is why the ancients held numbers in such high regard! It is because they are, in some sense, imperishable — they are not subject to change and decay, as are all the objects of common experience. So, they’re nearer to the “source of being”.’ This insight seemed both simple and obviously true, but I was to discover that it is neither, according to much contemporary philosophy.

I later found some validation of the basic intuition in a standard text on the subject (my bolds):

Neoplatonic mathematics is governed by a fundamental distinction which is indeed inherent in Greek science in general, but is here most strongly formulated. According to this distinction, one branch of mathematics participates in the contemplation of that which is in no way subject to change, or to becoming and passing away. This branch contemplates that which is always such as it is and which alone is capable of being known: for that which is known in the act of knowing, being a communicable and teachable possession, must be something that is once and for all fixed ~ Jacob Klein, Greek Mathematical Thought and the Origin of Algebra.

Reality and Existence

This insight lead me to ponder what it means to say that number and phenomenal objects exist in different ways. Until this time, it had never occured to me that there might be different ways of existing; I had thought that things either exist, or they don’t.

So: the word ‘exist’ is derived from a root meaning to ‘be apart’ or ‘separate’ — where ‘ex’ means apart from or outside (compare exile, external), and ‘ist’ mean to be or to stand (reference). Ex-ist, then, means to be a seperable object, to be ‘this’ as distinct from ‘that’. This applies to all the existing objects of perception — chairs, tables, stars, planets, and so on — everything which we would normally call ‘a thing’ or ‘a phenomenal object’.

But then, consider number. Obviously we all concur on what a number is, and mathematics is lawful; we can’t just make up our own numbers. But numbers don’t exist as do objects of perception; there is no object called ‘seven’². You might point at the numeral “7”— but that is a symbol. What we concur on, what it actually comprises, is a number of objects, but the number cannot be said to exist independently of its apprehension. A number is real as an act of counting or as an estimation of quantity (according to an approach to philosophy of maths called intuitionism). In any case, it is something that can only be known to a mind; it has no material existence.

But even if a number doesn’t exist in the same sense that tables and chairs do, it is indubitably real – real, that is, for anyone who is capable of counting. If I ask a grocer for a dozen eggs, and receive 11 or 13, I’ll know that something is up, to give a simple example; get your math wrong, and your Lunar Lander will crash, to give a rather more sophisticated one. As we’ve seen, the Greek philosophers observed that number has a higher degree of exactitude and determinateness than do ordinary sensory objects (hence the expression to know something ‘with mathematical certainty’).

But then, I wondered, in what domain or sense do numbers exist? ‘Where’ are numbers? How can they be real? Perhaps, came the thought, they exist in an intelligible domain, of which cognition is an irreducible part, and so, accessible only by reason. So from this I concluded that numbers are not existent in the same way that phenomenal objects are; that they are real in a way that is different to the mode of existence of phenomenal objects.

After thinking more about this, I realised I was contemplating the Platonic distinction between ‘intelligible objects’ and ‘objects of perception’. Objects of perception only exist, in the Platonic view, because they conform to, or are instances of, Forms or Ideas. Particulars are ephemeral instances of Forms, but they have no inherent reality, as they can (and do!) vanish or change with time (and so were not considered objects of true knowledge in Platonist philosophy). Phenomenal objects of perception exist, but only in a transitory and imperfect way, whereas the archetypal Forms are real and pre-existent in the fabric of the Cosmos: while they do not materially exist they provide the basis for existing things by creating the patterns and ratios against which things are formed. They are real, above and beyond the existence of wordly things; but they don’t actually exist. They don’t need to exist; things do the hard work of existence.

Here then, I thought, a distinction can be made between reality and existence. This seemed to me a major breakthrough³. It also introduced me to the idea of there being ‘degrees of reality’– that there are things that are more and less real, something which has generally dropped out of today’s philosophy.

This, of course, turned out to be Platonism 101 — which was puzzling, as I’d never really studied Plato in any depth, despite having completed two years of undergraduate philosophy⁴.

The Nature of Universals

Subsequently I was to discover that the reality of number is a sub-set of the broader question of the reality of universals⁵. Universals are the descriptive attributes or characteristics that are shared by different instances of particular beings; ‘whiteness’ or ‘being triangular’ are often given as examples. I learned that in medieval philosophy, there was a long-running debate between those who accepted the reality of universals (called ‘realists’ but meaning something very different to what we mean by that today) and those who claimed that such attributes were mere names (‘nominalists’) and that only particular things are real. This debate is central to the entire corpus of metaphysics and I learned that calling the reality of universals into question had the effect of unravelling the intricate tapestry of scholastic philosophy grounded in the Platonic-Aristotelian tradition⁶.

The question of universals is connected to the above distinction between sensory and intelligible objects. Sensory objects are situated in time and place and are detectable by the senses (or by the instruments which augment the senses). These are the basic furniture of the empirical world, comprising the totality of the visible and detectable Universe. Empiricist philosophy is grounded in the reality of objects of experience — things that can be known objectively and explained theoretically. Whereas the question of the reality of abstract objects is perplexing, because they’re not among those objects, even though (and also perplexingly!) we rely on mathematical reasoning in making sense of the objective domain.

So the objection is, according to an article What is Math, Smithsonian Magazine:

The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God’⁷

The essay goes on to say that to admit the reality of abstract objects means that ‘empiricism goes out the window’. As noted, the scientific worldview is grounded in empiricism, in the conviction that what is ‘really out there’ exhausts what is real. This deep conviction (not to say prejudice) colors almost every discussion about the question. I think this is because nominalism prevailed in the medieval debate history, and as is often the case, ‘history was written by the victors’. As has been argued, the abandonment of the belief in the transcendental reality of universals was a major step in the dissolution of metaphyics in Western culture.

So if the ‘reality of number’ seems perplexing, it is because the culture we inhabit has long since adopted nominalism, which holds that only particulars are real. It’s simply become the assumed consensus, what is nowadays ‘commonsense realism’ — and within it, there is no conceptual space for the reality of the immaterial⁸.

Bertrand Russell provides a perceptive insight into the nature of universals, in a chapter from one of his early texts (my emphases):

Suppose…that we are thinking of whiteness. Then in one sense it may be said that whiteness is ‘in our mind’. …[But] in the strict sense, it is not whiteness that is in our mind, but the act of thinking of whiteness. The connected ambiguity in the word ‘idea’ … also causes confusion here. In one sense of this word, namely the sense in which it denotes the object of an act of thought, whiteness is an ‘idea’. Hence, if the ambiguity is not guarded against, we may come to think that whiteness is an ‘idea’ in the other sense, i.e. an act of thought; and thus we come to think that whiteness is [only] mental. But in so thinking, we rob it of its essential quality of universality. One man’s act of thought is necessarily a different thing from another man’s; one man’s act of thought at one time is necessarily a different thing from the same man’s act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts.

This is the sense in which universals were what the medievals would designate as ‘incorporeal objects’. They are real, in that they’re the same for all who can grasp them, but they can only be known as ‘objects of reason’. They’re not materially existent, or ‘out there somewhere’ in an empirical sense. They’re in the mind, but not of it.

I later encountered a statement from Albert Einstein, who said:

I believe… that the Pythagorean theorem in geometry states something that is approximately true, independent of the existence of man.⁹

Perhaps something that Einstein overlooked is that while the Pythagorean Theorem might be true independently of the existence of human beings, it can only be grasped by a rational intellect, and to our knowledge, that is something only humans possess (leaving aside considerations of a ‘divine intellect’). In this sense, the Pythagorean Theorem is another example of an ‘incorporeal object’ — something real that can only be grasped by a rational mind.

It is well–known that Einstein fiercely defended a basic tenet of scientific realism, namely, the supposed ‘mind-independent’ nature of the objects of scientific analysis. His view was that the Universe simply is as it is, irrespective of whether us humans are aware of it or not, and it’s the task of science to understand it as it is. But this view doesn’t take into account the role of the scientist in the process of scientific discovery — he’s advocating what has since been criticized as ‘the view from nowhere’, the idea that science provides a perfectly objective viewpoint on the world as it truly is¹⁰. This was the major point of contention in Einstein’s long–running debate over quantum mechanics with physicist Neils Bohr, a principal architect of modern atomic theory, who insisted ‘It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about Nature.’

I think Bohr’s objection is cogent: he understands the sense in which mind is a factor, even in respect of the apparently mind-independent objects of scientific scrutiny. This, I believe, was driven home by the ‘observer problem’ in quantum mechanics, although that is not a topic we will explore further here.

The Ligatures of Reason

Insight into the reality of universals, I came to believe, was the golden key to understanding metaphysics. I formed the view that the rational intellect—which the ancient Greeks called ‘nous’ —maintains coherence by its grasp of such intelligible principles. This is a thread that goes right back to the origins of Greek philosophy and the debates on the Ideas from Parmenides and Plato. Deep waters, indeed, and subject to millenia of scholarly debate. But let’s consider a way in which it enters into our day-to-day understanding of the world.

Language itself, after all, depends on abstraction. Whenever we use the terms ‘same as’, ‘equal to’, ‘different from’, ‘less than’, and so on, we’re making use of our capacity for rational abstraction, usually without even being aware of so doing. This capacity is anticipated by a discussion in Plato’s Phaedo called ‘The Argument from Equality’. In it, Socrates argues that in order to judge the equal length of two like objects — two sticks, say, or two rocks — we must already have ‘the idea of equals’ present in our minds, otherwise we wouldn’t know how to go about comparing them; we must already have ‘the idea of equals’. And this idea must be innate, he says. It can’t be acquired by mere experience, but must have been present at birth.

I don’t know if it’s necessary for us to accept the implied belief in the ‘incarnation of the soul’ to make sense of the claim: the fact that it’s innate is what is at issue. It is the innate capacity which provides us the ability to make such judgements, which we as rational creatures do effortlessly.

On a larger scale, the same kind of capacities are brought to bear on formulating the mathematical bases of theoretical physics. Science sees the Universe through such mathematical hypotheses, which provide the indispensable framework for making judgements (in accordance with the oft-quoted Galilean expression that ‘the book of nature is written in the language of mathematics’).

Thus intellectual abstractions, the grasp of abstract relations and qualities, are quite literally the ligatures of reason — they are what binds rational conceptions together to form coherent ideas. In saying that I have, of course, wandered far from the original discussions in the Parmenides and the Phaedo, but I maintain that these themes at the origin of culture are unique to the heritage of Western philosophy and are still implicit in our ways of thinking, and that they are crucial elements in what gave rise to modern science, even if, due to its empiricist leanings, it has now lost sight of their significance¹¹.

But even though they’re real, they are not ‘out there somewhere’. ‘Perhaps not’, will come the objection, ‘but that’s because they’re simply products of the mind’. That is the sceptical view of much mainstream philosophy, by which it hopes to ground such principles in the physical domain, by depicting them as ‘produced by the brain’, shaped by evolution and, so, still the result of physical causes. While detailed discussion of this claim would be take us far afield of the main topic, suffice to note for the purposes of this essay that this attitude takes for granted the division of mind (‘in here’) and world (‘out there’) as being, to all intents, separate realities. And that itself is a metaphysical construction! Furthermore what this doesn’t account for, is the uncanny ability of mathematics and reasoning to discover novel facts about the nature of reality, which is at the basis of scientific discovery¹². It is also challenged by the emerging philosophical school of enactivism which assumes a deep continuity between life and mind¹³.

Conclusion: Universals as Structures in Consciousness

In an earlier essay I pointed to the way in which the brain~mind constructs our perceived world through the assimilation of sensory data and its synthesis with our conceptual systems and inherited knowledge. In that essay, I mentioned the book Mind and the Cosmic Order, by Charles Pinter, which provides a deep analysis of how the mind structures experience through the perception of ‘gestalts’, unified wholes. Pinter’s book is firmly grounded in evolutionary and cognitive science as well as being philosophically informed. He shows that what we percieve as objects in the external world are inextricably linked to the cognitive faculty which identifies gestalts as meaningful wholes set against a background. He shows that this process is operative in the cognition of even in very simple organisms¹³. His book provides considerable evidence for the kind of scientifically-informed idealism that I’m wishing to elaborate in my essays.

A crucial element of this perspective is the non-duality of self and world. The world is ‘the world as experienced’, the self is ‘self-in-the-world’. They are not two separate domains, but a single domain that has objective and subjective poles. This is also central to the approach of enactivism or embodied cognition as exemplified by the groundbreaking book The Embodied Mind (see bibliography), in which they draw considerably from Buddhist principles of non-duality.

In accordance with this approach, I’m contemplating the idea that universals, mathematical and logical principles and the like can be understood as ‘uniform structures within consciousness’. Among other things, they are ways in which intelligence organises experience. There are elements of that capability even in animals, as Pinter shows, but in human cognition, the faculty of reason greatly amplifies this capacity through the application of rational thought and language, and the ability to discern relationships and meaning. These structures are then encoded in cultural, mythic, scientific and linguistic patterns and conventions which are inherited through the generations. Within this cultural and intellectual medium, the ‘Ideas’, ‘Forms’ and ‘Universals’ represent something like the structure of possibilities: the forms that things must take in order to exist. As Kelly Ross says ‘ Universals exist precisely where possibilities exist’.

Portrayed in this light, the reality of universals is broadly compatible with a naturalist outlook, although probably can’t be accomodated within the procrustean bed of reductive materialism. Maybe we don’t see them because they’re too close to notice, woven, as they are, into the very ground of the language and reason that comprise our conscious experience. And that is because they are, after all, the ligatures of reason.

Footnotes

1. Later I realised the ‘indivisibility’ strictly speaking applies only to prime numbers, but the basic point remains.
2. It is true that numbers are often referred to as ‘abstract objects’, but I wonder if the use of the word ‘object’ in this context is somewhat metaphorical, or perhaps a façon de parler, as a number is not literally ‘an object’ although this is subject to debate.
3. I’ve since learned that C. S. Peirce distinguishes reality and existence along similar lines. For Peirce, something is real if it has intrinsic properties. Reality pertains to the way things are, regardless of opinion or belief. Existence, on the other hand, implies actuality in the physical world or in a specific context. To say something exists means that it is present in a given realm of discussion. For example, unicorns are a concept we can discuss and think about, but they don’t exist in the physical world. This allows him to conceive of God as real but not existent.
4. Platonism is stock-in-trade for those who studied ‘The Classics’ but these texts had long been dropped from the curriculum presented to me in both school and University (notwithstanding a perfunctory year of Latin in junior high school). I’ve read up on it since, but I’m aware of the many gaps in my education. I have since come to wonder whether my fascination with the subject is due to cultural inheritance — that these ideas are part of the cultural milieu in which I was born and raised, even if I was not consciously aware of them, which is why they resonate so very strongly with me. A Platonist — or a Buddhist! — might even ascribe it to past-life memories.
5. Universals and numbers are not the same concept, but there are overlaps, particularly with respect to their shared status as abstract objects.
6. For which, see The Theological Origins of Modernity, Michael Allen Gillespie, University of Chicago Press 2008.
7. What is Math? Smithsonian Magazine (Online, retrieved 24/09/23).
8. For this point, see Joshua Hochschild, What’s Wrong with Ockham? Reassessing the Role of Nominalism in the Dissolution of the West, (Online, retrieved 24/09/23) — in-depth account of the eclipse of realism and it’s pervasive effects on Western culture and philosophy.
9. When Einstein Met Tagore (Online, Magazine Article, original 1931, retrieved 24/09/23).
10. Thomas Nagel, The View from Nowhere (see bibliography).
11. Jacques Maritain: ‘‘What the Empiricist speaks of and describes as sense-knowledge is not exactly sense-knowledge, but sense-knowledge plus unconsciously introduced intellective ingredients — sense-knowledge in which s/he has made room for reason without recognizing it. A confusion which comes about all the more easily as, on the one hand, the senses are, in actual fact, more or less permeated with reason in humans, and, on the other, the merely sensory psychology of animals, especially of the higher vertebrates, goes very far in its own realm and imitates intellectual knowledge to a considerable extent.” (see bibliography)
12. See Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Science (online, retrieved 24/09/23)
13. See Varela, Thompson, Rosch — The Embodied Mind (refer to biblio)
14. ‘Everything you see, hear and think comes to you in structured wholes: When you read, you’re seeing a whole page even when you focus on one word or sentence. When someone speaks, you hear whole words and phrases, not individual bursts of sound. When you listen to music, you hear an ongoing melody, not just the note that is currently being played. Ongoing events enter your awareness as Gestalts, for the Gestalt is the natural unit of mental life. If you try to concentrate on a dot on this page, you will notice that you cannot help but see the context at the same time. Vision would be meaningless, and have no biological function, if people and animals saw anything less than integral scenes’ — Charles Pinter (see biblio)

Biblography

M.A Gillespie, The Theological Origins of Modernity, University of Chicago Press, 2008

Joshua Hochschild, What’s Wrong with Ockham? Reassessing the Role of Nominalism in the Dissolution of the West, (Online, retrieved 24/09/23)

Jacob Klein, Greek Mathematical Thought and the Origin of Algebra, Dover Publications; Revised ed. edition (September 11, 1992)

Jacques Maritain, The Cultural Impact of Empiricism (retreived 24/09/2023)

Thomas Nagel, The View from Nowhere, Oxford University Press; Revised ed. edition (9 February 1989)

Charles Pinter, Mind and the Cosmic Order, Springer International, 2021

Kelly Ross, Meaning and the Problem of Universals, (online, retrieved 24/09/23)

Bertrand Russell, The Problems of Philosophy, Chap. IX, The World of Universals, (online, retrieved 24/09/23)

Eleanor Stump & Norman Kretzman (ed.), The Cambridge Companion to Augustine, Cambridge University Press, 2001 — Chapter 6, The Divine Nature, On Intelligible Objects

Varela, Thompson, Rosch, The Embodied Mind, MIT Press Academic; 2nd Revised ed. edition (13 January 2017)

Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Science (online, retrieved 24/09/23)