Do Numbers Exist?

According to your disposition, you might have an immediate gut reaction to this question. My initial reaction (oh so long ago) was: “Of course numbers don’t exist. You can’t pick up the number 3 and throw it through a window.” That is, my intuition was that the only things that exist are the kinds of things that can be physically manipulated, and numbers, by almost every account, just aren’t this kind of thing.

To be clear about our terms, you can pick up numerals — that is, you can pick up concrete instances of numbers, like the plastic number signs at the gas station telling you how much gas costs, or the printed numerals in a book, denoting page numbers. But you don’t, by virtue of tearing out page three of a book and tossing it out a window, throw the number 3 out the window, any more than you throw me out of a window by drawing a picture of me and throwing that out the window.

Numbers, if they exist, are generally what philosophers call abstract objects, and those who maintain that such things exist claim that they exist outside of space and time. If you’re like me, you shake your head at such talk. “Outside of space and time? What does that even mean? Gibberish!” If you are similarly disposed, you might be a nominalist (in case you’re accumulating self-descriptive philosophical terms), and you are part of a long, proud philosophical tradition that thinks that existence is the exclusive domain of the physical.

However, your nominalism begins to run into problems pretty quickly. Never mind numbers. What about things like, say, novels? What exactly is the novel The Catcher in the Rye? It’s not any of the particular instantiations of it — it’s not the copy on your bookshelf; it’s not the copy on mine. All of the print copies on the planet could be eradicated and still the novel could be able to be said to exist. Is the novel the original manuscript sitting in a safe somewhere? But that could be burned and you could still argue that the novel exists. But if the novel itself is not identified with any of its particular instantiations, then the nominalist is in a bit of a quandary. On this perspective, the copies of the novel are instantiations of the novel itself, and the novel itself is seeming to be something abstract — something non-physical.

So the idea of something somehow existing outside space and time is suddenly not as absurd as it may have seemed. What about numbers, then? Of course there are disanalogies between numbers and novels. Novels are invented by humans, while, on most views of the subject, numbers exist whether or not humans ever happened to discover them. But, putting such differences aside for the moment, perhaps the existence of novels as abstract objects gives us some traction to say that numbers exist as abstract objects.

Abstract objects

What other sorts of things could be included in the category of abstract objects? The funny thing is that in many seminal texts on the subject, one has to plumb deep to find mention of what would count as an abstract object. Mathematical objects generally top the list (numbers, points, lines, triangles, etc.), followed by things like chess moves, games in general, pieces of music, and propositions. How are these things abstract? We generally think of a chess move, for instance, as something that exists by virtue of a concrete chess player actually moving a concrete chess piece in accordance with the rules of the game (which could themselves be considered abstract, but never mind this for the moment). But that seemingly concrete move can be instantiated in so many concrete ways — you could be replicating someone else’s game on your own chess board, you could make the move on a hundred different boards all at (nearly) the same time, you could make the move in your head before you make it on the board,… and all of these concrete possibilities point to the metaphysical problem here: If you believe there is only one move, and it’s concrete, then which move is the one move? And then what are the other moves? Copies of the move? Or instantiations of the same move? If you believe in abstract objects, you have, on some takes, an easier time of it. The move itself is an abstract object, and every physical version of that move is a concrete instantiation of that move. That is, none of the concrete, physical moves are actually the move — there is only one move and it is abstract, and any physical move is a copy, like a sculpture of a real person. (You can have a thousand sculptures of a person, but there’s only one person. The sculptures are imitations or instantiations of the person.)

This perspective is (loosely) called platonism, named after Plato’s idea that there are ideal “forms” — perfect archetypes of which objects in the real world are imperfect copies.

Why would these ideal forms not exist in space-time? I.e., why would they have to be abstract? Well, objects in space-time (the real world) are all imperfect copies of something. So if an ideal form existed in, say, your living room, then it would be non-ideal by virtue of existing in your living room. To put it perhaps less question-beggingly, if, say a chess move were instantiated in a thousand ways, how would you pick out the ideal version from which all others were copied? All of the instantiations would have similar properties, and so no one instantiation would stand out as different enough to count as the move, the platonic form of that move. Therefore, it makes sense to posit an abstract version of the move — something perfect, and outside of space-time, from which all the worldly versions are copied.

Thinking about geometric objects is perhaps the clearest way to think about abstract objects. A line segment (a true, geometric line segment) is a perfectly straight, one-dimensional object with a determinate length. There are no such objects in space-time. Every object you could possibly interact with is three-dimensional — no matter how thin a piece of, say, plastic you create, it always has a height and a thickness, giving it three dimensions. Nothing, therefore, in the concrete world, is a real geometric line segment. We have things that approximate line segments — very straight, very thin objects. But none of those things will ever be perfectly straight and with zero thickness. So if there does, somehow, exist a true line segment, it certainly isn’t in the concrete world, and therefore it must be in some sort of abstract realm.

Knowledge of abstract objects

One of the most damning aspects of platonism is its failure to come to terms with how we learn things about abstract objects. The general picture of how we acquire knowledge goes something like this: We perceive an object in the physical world, via physical means (e.g., light bounces off the physical object and hits our eyes), and eventually we process such perceptions in our brains and work with mental representations — i.e., brain states — of the object in question. But an abstract object can’t be processed like this. It is non-physical, and so, e.g., light can’t reflect off of it. So our usual causal theory of knowledge acquisition fails for things like numbers.

Well, then, how is it that we come across any knowledge of abstract objects, if they indeed exist? Some mathematical platonists, like the venerable logician Kurt Gödel, resorted to the idea that we just know truths about mathematical abstracta. As he wrote:

But, despite their remoteness from sense experience, we do have a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception…

But this is clearly an unacceptable answer to the problem of knowledge of abstract objects. How exactly do the axioms of set theory force themselves upon us? Waving your hands and saying “they just do” isn’t an account of the process, and leaves us in the dark as to how they just do, which is precisely what we need before we can take the platonist seriously as an epistemologist. (One need merely look at the history of geometry to see one serious problem with seeing the “obvious” truth of axioms. Until Lobachevsky and Riemann came along with consistent non-Euclidean geometries, nearly everyone though that Euclid’s fifth postulate “forced itself upon us”.) How does some feature of a non-spatiotemporal object force itself upon our spatiotemporal brains? The only way would be somewhat magical, and you could look to Descartes to see the folly of such a plan. Descartes posited that minds are distinct substances from brains, and indeed were non-spatiotemporally located. Of course, this leaves the problem of how the mind somehow slips into the brain and affects it. Descartes’ answer was that it crept in through the pineal gland. But this is no answer; it merely delays the answer for a moment. How does the non-spatiotemporal mind creep in through the pineal gland, and then into the brain? Descartes had no answer for this, of course, because the whole thing would be terribly mysterious, explaining how the non-physical interacts with the physical.

Worries like this keep nominalists well-motivated to stay on their side of the debate.

The argument from indispensability

Even if you’re dead-set against granting the existence of numbers, you think platonism is absurd, you have challenged platonism’s picture of knowledge, and you somehow have all of your nominalist ducks in a row, there is still one very influential argument to contend with as regards numbers’ existence: The argument from indispensability. Hardcore nominalists are often quite scientifically-minded, scientifically-motivated philosophers. And it is this love of science that gets them into trouble with denying the existence of numbers. The argument runs, in broad strokes, like this:

  1. Science is the best arbiter of what exists.
  2. Therefore, if science says something exists, we should accept it.
  3. Science relies (heavily and intractably) on mathematics.
  4. Therefore, science says that numbers exist.
  5. Therefore, numbers exist.

If you’re a good nominalist, you’re more than likely feeling obliged to accept this argument as sound. But if you accept its conclusion, then you’re right back to the issue of explaining what numbers are. They can’t be physical objects, therefore they must be abstract. But, as a nominalist you claim that there are no abstract objects! And you are caught in an intractable dilemma.

Many nominalists give up at this point. Hilary Putnam wrote resignedly:

Quantification over mathematical entities is indispensable for science…; but this commits us to accepting the existence of the mathematical entities in question. This type of argument stems, of course, from Quine, who has for years stressed both the indispensability of quantification over mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes.

The talk of “quantification” is a bit of logic talk, but we can paraphrase it into regular English: “If science uses numbers, then science is committed to the existence of numbers.” You might see a glimmer of nominalist hope here. Science also uses frictionless planes, for example, and yet no scientist feels committed to the existence of those. Perhaps there is a way out of our commitment to numbers in the same way. Or perhaps, one might argue, frictionless planes actually do exist as platonic, abstract objects.

But there are two more “obvious” ways to be a nominalist about mathematics.

First, you could argue that numbers exist, and are actually physical objects. Penelope Maddy argues something close to this in her early work, Realism in Mathematics. She actually is here arguing for a version of naturalized platonism — the idea being that what is usually thought of as abstract objects are actually somehow existent in the physical world. But, platonist labels aside, the gain for nominalism on this take would be obvious: numbers, if they are physical objects, would be just another part of the down-to-earth nominalist physical world, like cats, trees, and quarks. This brave strategy, however, ultimately fails. It would take us into some metaphysical thickets to explain why, so I have relegated this to a paragraph at the very end of this post.

Second, you could argue that numbers aren’t actually indispensable to science. Hartry Field famously tried this strategy, claiming that science in fact only seems to rely on mathematics. On Field’s view, this seeming reliance is really just a fiction. In order to prove this Field attempted to nominalize a chunk of physics, by removing all reference to numbers within it. This complicated, counterintuitive project has met with equal parts awe and criticism. The consensus is that his project is untenable in the long term.

So do numbers exist or not?

Well, if you’re a platonist, you would answer “yes, numbers exist”. And further you would claim that they possess a sort of existence that is abstract — different from the sort of existence that stones, trees, and quarks enjoy. Of course, this means you are in the unenviable position of explaining the coherence of this sort of existence, along with the herculean task of explaining how we know about anything in this abstract, non-physical realm.

If you’re a nominalist, you’d probably answer “no, numbers do not exist”. However, now you have the unenviable job of explaining why mathematics seems so indispensable to science, while science is perhaps our best tool for saying which things exist. The two best nominalist answers to this conundrum seem untenable.

Probably, as is usually the case in philosophy, dogmatically sticking to one side of a two-sided debate will be inadequate. Maddy’s attempt at naturalizing platonism was a brave bridge across the nominalist-platonist divide, but clearly isn’t the right bridge. We’ll examine some other options in a future post.


References and Further Reading

Balaguer, Mark. (1998) Platonism and Anti-platonism in Mathematics. Oxford: Oxford University Press.

Benacerraf, Paul. (1973) “Mathematical Truth”, Journal of Philosophy 70.

Colyvan, Mark. (2001) The Indispensability of Mathematics. Oxford: Oxford University Press.

Irvine, A.D. (1990) Editor. Physicalism in Mathematics. Dordrecht: Kluwer.

Lowe, E. & Zalta, E. (1995) “Naturalized Platonism Versus Platonized Naturalism,” Journal of Philosophy 92.

Maddy, Penelope. (1992) Realism in Mathematics. Oxford: Clarendon Press. Revised paperback edition.


A note on Maddy’s naturalized platonism

Maddy actually thinks that we perceive sets. Number theory, as many logicians are proud to point out, can be reduced to set theory — i.e., numbers can be reduced to sets, which are, of course, generally seen as just another sort of abstract object. Maddy’s move is to bring those sets into the natural world. So that when we see an egg, we are perceiving that egg, but are also perceiving the set containing that egg. (A set containing an object is different from the object itself, you may recall from your math studies.) And that set containing the egg is a natural object, different from the egg itself. But now we run into trouble. Certainly there must be something different between an egg and a set containing that egg; otherwise ‘set containing that egg’ is just a proper name denoting the egg in question, and nothing metaphysical hangs on the distinction. (If you call me “Alec” or “author of this post”, you are not positing the existence of two people — these are just two different names for the same person.) Well, the usual distinguishing feature of abstracta is that they are not spatiotemporally located; but on Maddy’s scheme sets are spatial objects. The problem: Our egg and the set containing it necessarily co-exist in the same exact region of space-time, and yet they are supposed to be different things. In what does this difference consist? Well, certainly nothing physical, otherwise they wouldn’t co-exist in the exact same region of space-time. But then the difference must be something non-physical — i.e., something about the set must be abstract. And if this is the case, then we’re right back to all of the problems inherent in platonism, particularly the problem of how we can have any knowledge of such abstracta.

On Definitions in Philosophy

When trying to define a term, we think generally of providing a set of necessary and sufficient conditions: a recipe for including or excluding a thing in a particular category of existence. For instance, an even number (definitions tend to work best in the mathematical arena, since definitions there can be as precise as possible) is definable as an integer that when divided by 2 does not leave a remainder. It is easy, given this definition, to ascertain whether or not a given number is even. Divide it by two and see if it leaves a remainder. If it does, then it’s not even; if it doesn’t, then it is. We have here a clear test for inclusion or exclusion in the set of even numbers.

Outside of mathematics, things get trickier. (Inside mathematics, things can be tricky as well. Imre Lakatos‘ excellent book Proofs and Refutations details some of the problems here. If you are mathematically and philosophically inclined, this is a must-read book.)

In Ludwig Wittgenstein‘s Philosophical Investigations, he famously talks about the travails of defining the term “game”. Is there a set of necessary and sufficient criteria that will let us neatly split the world into games and non-games? For instance, do all games have pieces? (No, only board games have these.) Winners and losers? (There are no winners in a game of catch.) Strategy? (Ring-around-the-rosie has no strategy.) Players? (Well, since games are a particularly human endeavor, it would be an odd game that had no human participants. But, of course, some games have only one player.) There seems to be no single set of characteristics that spans across everything we’d like to call a game. Wittgenstein’s solution was to say that games share a “family resemblance” — “a complicated network of similarities overlapping and criss-crossing”. A great many games have winners and losers, and so share this family trait; and then there are games that have pieces, and this is another trait that can be shared. Many (but not all) of the games with pieces also have winners and losers, and so there is significant overlap here. Games with strategy span another vast swath of the game landscape, and many of these games have winners and loses, many of which also have pieces. But not all. And so a networks of resemblances between games is found — not a single boundary that separates games from non-games, but a set of sets that is overlapping and more or less tightly connected.

This is a brilliant idea, but one that often leaves analytical philosophers with a bad taste in their mouths. If you try to formalize family resemblances (and analytical philosophers love to formalize things), you run up against the same problems as you had with more straightforward definitions. Where exactly do you draw the line in including or excluding a resemblance? Games are often amusing, for instance. But so are jokes. So jokes share one resemblance with games. But jokes are often mean-spirited. And so are many dictators. And dictators are often ruthless. As are assassins. So now we have a group of overlapping resemblances that bridges games to assassins. And if you want to detail the conditions under which this bridge should not take us from one group of things (games) to the other (assassins), you are back to specifying necessary and sufficient conditions.

Wittgenstein, I imagine, would have laughed at this “problem”, telling us that we just have to live with the vague boundaries of things. Which is all well and good, but is easier said than done.

Knowledge

The defining of knowledge gives us a great example of definitions at work and their problems. For those of you who haven’t been indoctrinated in the workings of epistemology, it turns out that a good working definition for knowledge is that it is justified true belief.

Is Knowledge Justified True Belief
I take it as axiomatic as can be that something has to be believed to be known. If you have a red car but you don’t believe that it’s red, you don’t have knowledge of that fact. But, clearly, belief isn’t sufficient to define something as knowledge. For instance, if I believe that my red car is actually blue, I still don’t have any knowledge of its actual color. So we have to bring truth into the picture. If I believe that my car is red, and it is actually red, I’m certainly closer to having a bit of knowledge. But, again, this isn’t sufficient. What if my wife has bought me a red car that I haven’t seen yet. I believe it’s red because I had a dream about a red car last night. Do I have knowledge of my car’s color? I’d say not. We need a third component: Justification. If I believe that my new red car is indeed red because I’ve seen it with my own eyes (or analyzed it with a spectrometer, if the worry of optical illusions bugs you), then we should be able to say I do indeed have a bit of knowledge here.

In 1963, Edmund Gettier came up with a clever problem for this definition — one that presents a belief that is justified and true, but turns out to not be knowledge. Here is the scenario:

  • Smith and Jones work together at a large corporation and are both up for a big promotion.
  • Smith believes that Jones will get the promotion.
  • Smith has been told by the president of the corporation that Jones will get the promotion.
  • Smith has counted the number of coins in Jones’ pocket, and there are 10.

The following statement is justified:

(A) Jones will get the promotion and Jones has 10 coins in his pocket.

Then this statement follows logically (and is therefore also justified):

(B) The person who will get the promotion has 10 coins in his pocket.

But it turns out that the president is overruled by the board, and Smith, unbeknownst to himself, is actually the one will be promoted. It also turns out that Smith, coincidentally, has 10 coins in his pocket. Thus, (B) is still true, it’s justified, and it is believed by Smith. However, Smith doesn’t have knowledge that he himself is going to get promoted, so clearly something has gone wrong. Justification, truth, and belief, as criteria of knowledge, let an example of non-knowledge slip into the definitional circle, masquerading as knowledge.

More Games

Let’s get back to the problem of defining games, and say that, contrary to Wittgenstein, you’re sure you can come up with a good set of necessary and sufficient conditions. You notice from our previous list of possible necessary traits that games certainly have to have players. Let’s call them participants, since “player” is something of a loaded word here (a player presupposes a game, in a way). And now you also take a stand that all games have pieces. Board games have obvious pieces, but so, you say, do other games. Even a game of tag has objects that you utilize in order to move the game along. (In this case, you’re thinking of the players’ actual hands.) So let’s add that to the list, but let’s call it what it is: not pieces so much as tools or implements. And perhaps you are also convinced that all games, even games of catch, have rules. Some are just more implicit and less well-defined than others. So let’s stop here, and see where we are. We have participants, implements, and rules.

And now we begin to see the problem. If we leave it at that, our definition is so loose as to allow under the game umbrella many things that aren’t actually games. A group of lab technicians analyzing DNA could fall under the conditions of having participants, implements, and rules. But if we tighten up the definition, we run the risk of excluding real cases from being called games. For instance, if we tighten the definition to exclude our lab workers from the fun by saying that games also have to have winners and losers we immediately rule out as games activities like catch and ring-around-the-rosie.

Lakatos coined two brilliant phrases for these definitional tightenings and loosenings: “monster-barring” and “concept-stretching”. Monster-barring is an applicable strategy when your definition allows something repugnant into the category in question. You have two options as a monster-barrer: do your utmost to show how the monster doesn’t really satisfy your necessary and sufficient conditions, or tweak your definition to keep the monster out.

Concept-stretching allows one to take a definition and run wild with it, applying it to all sorts of odd cases one might not have previously thought to. For instance, perhaps we should expand entry into the realm of games to include our intrepid DNA lab workers. What would that mean for our ontologies? And what would it mean for people who analyze games? And for lab technicians?

Philosophers love to define terms; they also love to find examples that render definitions problematic. It’s a trick of the trade and a hazard of the business.

What is Philosophy?

What is philosophy? And why are we bothering to blog about it?

Even people trained in philosophy are often hard-pressed to come up with a pithy definition of it. The first time I taught Introduction to Philosophy, I stammered at the front of the class for a good five minutes trying to explain the sorts of things about which I was going to teach them for the next fifteen weeks. (Later in the semester, I stammered for significantly less time, but with the same significant stammering intensity, over the definition of “ethics”. So it’s not just the general term “philosophy” that’s the issue, I think.)

Of course, if you’re a philosophy aficionado you might well already know the problems attached to the process of defining terms. Wittgenstein, famously, in his Philosophical Investigations, took his readers down the rabbit hole in attempting to define the term “game” — even something so seemingly simple can be difficult to pin down with authority and without counterexamples getting in your way.

But it’s not just the general problem of defining terms that is difficult in the case of “philosophy”. The field to which the term attaches is so broad and so nebulous that it’s no wonder it’s so hard to describe.

It may be fruitful here to think of philosophy as a practice, rather than a field. And you learn about a practice (and how to participate in that practice) more by immersion than by definition. So, while it’s fairly unsatisfying to someone just starting out in the practice of philosophy, I think it’s actually not unfair to say at the beginning of a philosophy course “you’ll see what philosophy is by the end of the semester. For now, crack open your Descartes text and let’s talk…”

That doesn’t help you, our much-appreciated reader, to figure out what it is this blog is about, and whether or not you’ll still be a reader next week. So, despite my trepidation, let me take a stab at saying what philosophy is and why we’ll be blogging about it.

The roots of the word “philosophy” harken back to “lover of wisdom”. Indeed, philosophy is all about the love of knowledge, and unearthing pieces of knowledge wherever you can. And when I say “wherever” I’m not kidding. There are philosophical treatises on such abstruse topics as nonexistent objects, and on subjects as far ranging as everything from humor to subatomic physics.

What, you might ask, makes some bit of knowledge about subatomic physics a piece of philosophy rather than a piece of physics? There have actually been scientists who have argued that philosophy of science is about as useful as astrology; and even the great philosopher Bertrand Russell wrote: “as soon as definite knowledge concerning any subject becomes possible, this subject ceases to be called philosophy, and becomes a separate science.” (Bertrand Russell (1912). The Problems of Philosophy. New York: Henry Holt & Co.) His thought was that the sciences provide definite knowledge, while philosophy provides insightful burrowing into ideas that may someday become science. (Of course, Russell was writing in the heady days when it seemed as if science and mathematics would explain everything, but that’s another story…)

Whether or not that’s true, it is most assuredly true that the philosophy of science has hit upon and explored many important areas of knowledge that scientists, busy doing the important work they’re doing, might never have explored. The importance of exploring these areas remains an open question, but if you have a philosophical disposition, you would seldom if ever doubt the importance of what you were studying. Not because your area of exploration might yield anything, say, scientifically fruitful, but simply because if it’s an avenue of knowledge, whatever lays at the end of it, you want to go down that path. It’s the journey itself that is as important as what you find, along with the fact that whatever you’ve found, it was something that needed discovering.

I remember my first day as a graduate philosophy student, going to the library and just wandering down the aisles. At first, I stuck to the philosophy stacks, marveling at the breadth and depth of the tomes there. But eventually I wandered into the math stacks — a second academic love of mine — and spent some quality time there, once again marveling at the results of humankind’s curiosity. Then it was off to the psychology stacks, and the science stacks, and before I knew it, somehow I was in an aisle of books devoted to 18th Century England. I grabbed a book at random and read a chapter on witchcraft and its relation to the social norms of the times, and marveled at it, even though it was not really something I’d normally be interested in — someone had trodden down this path with great intellectual fervor, and had unearthed theories, knowledge, and connections that no one else had ever thought about in quite the same way. Before putting the book back, I noticed that no one had checked it out of the library for decades. This made me melancholy for a moment, until I realized that if I had written this book, though I’d certainly want people to read it, there would be a big part of me that would be content to have done the work and written it, regardless of my future audience. At least it was in a respected library, filling in a nook in our intellectual history.

If this story resonates with you, you might have a philosophical disposition.

Have I explained what philosophy is yet? Not really, I suppose; though I believe I have explained why we’re bothering to blog about it.

So what is philosophy? The first thing to keep in mind is that it’s always “philosophy of X”, where X can be just about any field. So we have philosophy of existence (generally called metaphysics), philosophy of knowledge (or epistemology), philosophy of morality (ethics), philosophy of art (aesthetics), philosophy of science, philosophy of mathematics, philosophy of language, philosophy of mind, philosophy of humor, philosophy of law, and so many other philosophies-of that it could make your head spin.

I was recently browsing for provocative philosophy paper titles (I thought it would be instructive to look at such titles in order to start to get a sense of what it is that philosophers do), and I came across this essay by Karel Lambert from back in 1974: “Impossible Objects”.

I haven’t read the article (come to think of it, I have to add that to my to-read list!), but I’m guessing that it’s a piece about such “things” as round squares. So now put yourself in a philosopher’s mindset for a moment. Someone says offhandedly to you: “why that’s as likely as a round square,” and you start thinking about that idea. A round square. Well, that’s impossible — such things couldn’t possibly exist. And this gets you thinking… there are things that don’t exist but could if the circumstances were right. Things like a 200-story building in Jamaica or six-legged cows. So there are two classes of things that don’t exist: possible (mammoth buildings in Jamaica) and impossible (round squares). Now you’ve begun carving up reality into interesting categories, and this is a particularly philosophical endeavor.

But wait… “Things” that don’t exist??? How could a thing be nonexistent? Is this really a problem of existence or just a trick of words? This well trod path leads one into the philosophy of language, where we ponder sentences like “The round square doesn’t exist.” Is this sentence true? Does “round square” refer to something in the same way that “George Washington” refers to something in the sentence “George Washington existed.”? These are very philosophical questions as well.

To be interested in why no skyscrapers exist in Jamaica is to be (probably) some sort of historian, sociologist, economist, or architect. To be interested in the difference between non-existent Jamaican skyscrapers and non-existent round squares, well, that’s being a philosopher.

See you next time. (?)