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One can ask whether the number one exists, and there are a range of answers. In particular, Platonism holds that this number does exist in some abstract world.

Now observe that number and one are words.

But one might also ask whether the word dog exists. For it to exist it must subsist somewhere. The natural place would be in Platonic Heaven, which has at least in contemporary discourse has been narrowed to include only things of a mathematical nature. It is only in Platos notion of Form that we see that it could exist, somewhere.

So why this turn away from Platos heaven, this narrowing of the gates, letting through only the plenitude of number in all its manifest multiplicity.

Why does one worry about the existence of the number one, but not that of the word dog? Why has the latter been fore-closed?

  • A dog is a concept but also has physical instantiations. A number is a concept but does not have physical instantiations. (You can't have 3 in the physical world, only 3 of something). Isn't that the essential difference? – user4894 May 26 '14 at 20:01
  • Sure, and this is a common angle of attack which points to a distinction between the two, and why perhaps the question of existence is sharper for numbers; but lets examine this supposition a little: in categorical foundations, a number can be any instantiation, ie 3 houses, 3 ideas, 3 letter 'A's etc. Its only when this notion is decategorified that we return to the first picture. That is 2 trees goes to just 2. If we go along with this foundation rather than conventional set theory the objection you raise appears to vanish, – Mozibur Ullah May 26 '14 at 20:18
  • @Mozibur_Ullah Can you tell me what you mean by "in categorical foundations, a number can be any instantiation, ie 3 houses, 3 ideas, 3 letter 'A's etc. Its only when this notion is decategorified that we return to the first picture ..." What are categorical foundations? My only point of reference would be using category theory as the foundation of mathematics. Clearly you mean something else. What should I be reading to understand your point? Categorification has a specific meaning in mathematics, clearly not what you are talking about. – user4894 May 26 '14 at 22:42
  • As you say yourself, we are worried about the existence of words. The existence of arbitrary (finite) words is not very different from the existence of arbitrary (natural) numbers. – Thomas Klimpel May 26 '14 at 23:08
  • The number one is far more useful to people than the word dog. Not that the word dog isn't important, it's important to many people, but the number one is important to almost everyone. Understanding the idea behind number one is important because understanding arithmetic/math is a survival trait, and these days it's a more important survival trait than understanding the word dog. – obelia May 27 '14 at 5:37
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I would speculate that it has to do with the (apparent) indispensability of numbers.

To unpack that a bit, I want to begin by talking about what it would mean for the word "dog" not to exist. Taking the word "dog" as a sign, let's divide it into signifier and signified, taking care to identify both as essentially abstract. What I mean by that is that the signifier "dog" is not the sound my lips make when I say "dog," nor the images of the letters 'd,' 'o,' 'g' on the screen before you, but the sequence of phonemes /d/ /o/ /g/ within the abstract structure of the English language. And likewise, the signified of "dog" is not any physical dog in the world, but the abstract concept of dog. So if we say that the word "dog" doesn't exist, then we're either saying the signifier or the signified of "dog" doesn't exist.

The existence of signifiers; the existence of concepts

Now, I think that to say that the signifier "dog" doesn't exist is tantamount to saying that some number doesn't exist! That's because if numbers exist, then sequences of discrete symbols exist; and so if sequences of discrete symbols don't exist, then numbers don't exist. So in that case, by worrying about whether the signifier "dog" really exists, then we're already worrying about whether numbers really exist. I think these two worries are close enough that they come and go together.1

That means that this question is really about the signified -- the concept of "dog." (This justifies Mauro's move towards concepts in his answer.) Now, obviously it would be wrong to say that the concept "dog" itself doesn't exist. It plainly does! We use the word dog, we think about dogs, we think of things as dogs, and so on. So it wouldn't make sense to worry about that. If we worry that the concept "dog" doesn't exist, then our worry must be that the concept "dog" is in some sense just made up -- that it's just in our heads, and it doesn't correspond to anything real. In that case, we could just skip past the concept, which is just made up, and go straight to the set of actual, physical dogs-in-the-world. That would make the concept "dog" a mere shortcut for referring to those dogs. This is the nominalist position.2 So the fundamental question you're asking is why it's harder to accept nominalism about numbers than it is to accept nominalism about dogs.

The problem with nominalism about numbers

A tempting answer is to say that it's obvious that we have dogs here before us, and so we can easily throw out the concept and just focus on the physical dogs that confront us. But we can't do that so easily with numbers; it's not obvious that any physical thing in the world confronts us with fiveness. It's true that we can find objects that come in fives, but have we really found fiveness in those objects, or have we structured our understanding of those objects using the concept five? As soon as we conceive them as discrete objects, we've given them a number. Couldn't we just as easily see five pebbles as ten half-pebbles? So it seems that the fiveness isn't in those objects after all.

Now, we could say the same thing about dogness -- dogness isn't really in dogs. After all, we don't have to divide the animal world up into species. There are many other coherent ways to construct groupings of animals that would help us to refer to the particular animals in the world. And we don't even have to use groupings of animals; we could think about groupings of ecosystems, or we could be reductionists and talk only about subatomic particles.

But note the difference between these two moves. When we throw out the concept "dog," we can use any number of other kinds of concepts to replace it with. But when we throw out the concept five... we can only replace it with the concept of another number! In our description of pebbles, we could do away with the concept of five only by introducing the concept of ten. There seems to be no way to do away with numbers entirely.

I should mention, finally, that this line of reasoning is (I think) closely related to the Quine-Putnam indispensability thesis, which I will playfully summarize as the argument that because we worry about the existence of numbers, we ought to go ahead and believe that they exist! However, there's a very interesting book called Science Without Numbers that attempts to show that numbers might not be indispensable after all -- hence the "(apparent)" in the first sentence.

1. Note that by worrying about whether numbers really exist, we're not necessarily worrying about whether the signifier "dog" really exists, because numbers have additional structure; they can be added, multiplied, and so on, while signifiers can't be. But if we say the signifier "dog" really exists, then we might as well go ahead and add the properties to "dog" necessary to treat it as a number, because we've already committed ourselves to the existence of an abstract object. So when we wonder whether numbers exist, we might as well be wondering whether the signifier "dog" exists; I think there's no good reason to reject the existence of numbers while accepting the existence of signifiers.

2. This line of reasoning applies to signifiers as well; when we worry about the existence of signifiers, we aren't worrying that they don't exist in our heads -- they obviously do! We're worrying about whether we've just made them up or whether they have some real correspondence to something outside our heads other than all of the physical "dog" sounds we make with our mouths and physical "dog" shapes that we draw and type. But again, in this case, it's clear that this worry is almost the same worry as that about the existence of numbers.

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  • I've always felt that a sufficiently sophisticated understanding of a scientific concept dispenses with quantity: it is understanding that counts, and that is qualitative. Quantity then becomes the expression of the qualitative. – Mozibur Ullah May 28 '14 at 23:39
  • Yes I'm sympathetic with that view. I clarified the above so that it doesn't sound like a defense of indispensability, which is not what I intended. – senderle May 29 '14 at 2:45
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I think that it is misleading to speak of "words". Words obviously exists.

According to me, we have to speak of "concepts".

Trying to "compress" a Treatise of Ontology in a few lines, we have objects; we are used to think at them as something that we can "see and touch" : the table, my keybord, the dog in the street.

Obviously, we have no problem in saying that such objects exists.

But there is another "type" of "entities" with which we interact : numbers, Higgs' boson, Google, society, ...

What they are ? It is hard to say that they are object like the above; at the same time, we "intercat" with them. We cam measure some physical magnitude connected with Higgs' boson. We "line in" a society with his laws. We have a continuous "dialogue" with Google.

Of course - and this is my preferred example - we can use numbers to count (the pencils on my desk, to manage my bank account, to design a bridge, ...

I persoanlly do not like the "platonic heaven", and I'm not able to develop a teneable account of abstract objects.

I prefer to call them concepts. Are they "physical" objects ? I think not. Are they only "fictions" ? If we do not like the hypotheses of a "collective dream", it is hard to say that "something" with which most of us "interact" quite continuosly is deprived of some sort of ... reality or existence.

In conclusion, dogs and pencils exists : I see some of them now.

I think that also concepts exists : in this precise moment in time I'm intercating with a web-site called philosophy.stackexchange.

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  • The wor dog designates a genus, a particular dog belogs to that genus. Can wenot think if the number 3 as a genus, and a specific 3 bottles belongs to that genus? – Mozibur Ullah May 27 '14 at 8:15
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    @MoziburUllah - I understand your example, but I'm less interested into "universal" (or Platonic forms : the "doggity" common to all dogs) than to other "intangible" entities, like numbers and Google. Is "the" number three a platonic form which is "incarnated" into all "triple" ? I think not. I prefer Frege's account: for sure numbers are concepts (second level one, i.e. "property" of concepts instantiated by e.g."triples"); are they also "abstract" objects (as Frege believed)? I do not know. – Mauro ALLEGRANZA May 27 '14 at 8:20

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