The German mathematician, logician and part-time philosopher Gottlob Frege (1848-1925) wasn’t well-known during his lifetime, but he’s now considered the father of analytic philosophy, the type of philosophy most professors in English-speaking countries and Scandinavia do. (Being the father of analytic philosophy makes you very well-known in certain circles.)
In 1918, Frege published an article called “The Thought” (Der Gedanke), in which he drew an interesting distinction. In addition to the standard categories of the mental and the physical, Frege said that “a third realm must be recognized”. This is the realm of meaning or sense that’s independent of anyone’s particular ideas.
First, therefore, is the realm of spatiotemporal things (“such as trees, stones and houses”). Then there is the realm of particular ideas in specific people’s minds (“an inner world distinct from the outer world, a world of sense impressions, of creations of [the] imagination, of sensations, of feelings and moods”). Lastly, there is the realm of what Frege called “thought”. The occupants of this third realm are similar to ideas, in that they “cannot be perceived by the senses”, but they are also similar to things, in that they “need no bearer”, i.e. they need not exist in anyone’s mind.
Thus, the thought [expressed by the Pythagorean theorem, for example] is timelessly true, true independently of whether anyone takes it to be true. It needs no bearer. It is not true for the first time when it is discovered….
It’s only because there is a single, commonly accessible thought that expresses the Pythagorean theorem that each of us can refer to it (the identical theorem) and agree or disagree about its truth value. Otherwise, my Pythagorean theorem would differ from yours, since the particular ideas in my mind are always and necessarily my ideas and never yours (and vice versa).
Indeed, the fact that we are able to use language to agree or disagree about particular propositions is evidence that this third realm exists:
If it is not the same thought … which is taken to be the content of the Pythagorean theorem by me and by another person, one should not really say “the Pythagorean theorem” but “my Pythagorean theorem, “his Pythagorean theorem”, and these would be different.
According to Frege, the thoughts that inhabit this third realm aren’t all propositions of mathematics or logic. He asks us to consider a statement like “This tree had green leaves”. Once we specify a time — “This tree had green leaves on July 1, 2004” — we have a statement that expresses a thought, which “if it is true, is true not only today or tomorrow but timelessly”.
Personally, I don’t find Frege’s notion of a third realm terribly convincing. I think there’s only one realm that has anything in it. But, as a metaphor, the “third realm” captures something extremely important. Along the same lines, the philosopher Charlie Huenemann recently began a post called “Reality Is Down the Hall” by quoting Schopenhauer:
“It is therefore worth noting and indeed wonderful to see, how man, besides his life in the concrete, always lives a second life in the abstract.”
Schopenhauer’s “second life” has this in common with Frege’s “third realm”: they both evoke what philosophers now call “abstract entities” or “abstract objects”:
Thus it is universally acknowledged that numbers and the other objects of pure mathematics are abstract (if they exist), whereas rocks and trees and human beings are concrete. Some clear cases of abstracta are classes, propositions, concepts, the letter ‘A’, and Dante’s Inferno. Some clear cases of concreta are stars, protons, electromagnetic fields, the chalk tokens of the letter ‘A’ written on a certain blackboard, and James Joyce’s copy of Dante’s Inferno. [Stanford Encyclopedia of Philosophy]
One world is at our fingertips, at the tips of our tongues, and folded into our fields of vision. The concrete world is just the world; and the more we try to describe it, the more we fail, as the here and now is immeasurably more vivid than the words “here” and “now” could ever suggest….
The second world is the one we encounter just as soon as we begin thinking and talking about the here and now. It is such stuff as dreams are made on; its substance is concept, theory, relation.
He then describes how we construct models of the concrete world, in particular, how scientists construct models that are increasingly intricate. He quotes Sir Arthur Eddington’s famous description of two tables:
[One table] has extension; it is comparatively permanent; it is coloured; above all it is substantial. By substantial I do not merely mean that it does not collapse when I lean upon it; I mean that it is constituted of “substance” and by that word I am trying to convey to you some conception of its intrinsic nature. It is a thing; not like space, which is a mere negation; nor like time, which is – Heaven knows what! … I do not think substantiality can be described better than by saying that it is the kind of nature exemplified by an ordinary table….
My scientific table is mostly emptiness. Sparsely scattered in that emptiness are numerous electric charges rushing about with great speed; but their combined bulk amounts to less than a billionth of the bulk of the table itself. Notwithstanding its strange construction it turns out to be an entirely efficient table. It supports my writing paper as satisfactorily as table No. 1; for when I lay the paper on it the little electric particles with their headlong speed keep on hitting the underside, so that the paper is maintained in shuttlecock fashion at a nearly steady level. If I lean upon this table I shall not go through; or, to be strictly accurate, the chance of my scientific elbow going through my scientific table is so excessively small that it can be neglected in practical life.
Huenemann points out that, “as educated beings”, we accept that the table described by science is “ultimately the real one”. He writes that the “second table somehow gives rise to the first table”, the one that seems perfectly solid from our human perspective. (At least, those of us who are “scientific realists” accept the ultimate reality of the second table, unlike those sometimes called “instrumentalists” who think the theoretical entities of science are merely useful devices for coping with the world.)
But then Hueneman seems to question our belief in the reality of the second, scientific table:
…here is an odd inversion – the first table, the one in the concrete world, is not quite as fully real as the abstract and dreamy second table, the one we never actually see, the one that is supposed to be a swarm of charged gnats, or packets of probabilities. The concrete table turns out to be an illusion. It arises somehow from the abstract world as does a mirage from heat and the bending of light. Isn’t that remarkable? Our official policy is to take the abstract to be more real than the concrete.
But, of course, the second, scientific table isn’t abstract or dreamy at all. It’s true that we can’t perceive it, but if the physicists are correct, it’s completely concrete. Instead, it’s the physicists’ description of the table, the theory of atoms and electrons, that’s abstract. Using Frege’s terminology, the meaningful propositions that scientifically describe the table belong to the third realm; the subatomic particles in the table, the marks in physics textbooks and the table in your dining room are concrete and belong to the first realm; and the particular ideas you and I have in our minds about such things belong to the second realm. It’s only the third realm of meaning or sense that is abstract. It’s abstract but, according to Frege, very real, even though its reality “is of quite a different kind than that of things”.
In practical terms, of course, it makes little difference whether we say abstract objects like numbers and propositions exist or not. Nobody thinks they float around in some ethereal, non-spatiotemporal realm (it would be quite impressive if they did). However, the fact that we can think about such things, real or not, is obviously one of the characteristics that makes humanity special, maybe the only thing that makes us special.
Furthermore, we act as if abstract objects were real, treating them with the utmost respect. Where would we be without the number 7, for example? Or 3 or 19, for that matter? Where would you be without the abstract object that is your name? And how about the Golden Rule? Or the concepts of truth or justice? As Schopenhauer said, it’s remarkably wonderful that we live with such things.