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The number 1 is used in language when we make claims of existence concerning distinct well-defined objects. It seems then that to say the number 1 does not exist would imply that nothing exists at all. Is this reasoning faulty? If I need a concept whenever I make positive claims would this imply that the concept actually exists, or is it possible that a concept can have a practical purpose, being used to make positive claims in philosophy, and at the same time not exist.

Also, does the quantifier there exists imply there exists exactly one?

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    I am asking a more specific question. There seems to be something unique about the number 1 as it pertains to our use in language. I see the word "a" very much like the phrase "to be." – REX Jul 24 '11 at 16:13
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    So, I completely changed my question; however, my question is still related to the ontology of numbers. However, I am interested in a specific type of reasoning rather than the question about the ontology of the natural numbers. – REX Jul 24 '11 at 16:38
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    Thank you, REX. I would still encourage any clarifying reformulations you may wish to make, but that said I am voting to reopen. – Joseph Weissman♦ Jul 24 '11 at 16:39
  • You may find the lexicon of the amazonian Piraha tribe of interest en.wikipedia.org/wiki/… . To the Piraha, something existing would imply at best only that a small quantity of that thing existed. – Tom Boardman Jul 24 '11 at 18:13
  • I am vaguely aware of this tribe. – REX Jul 24 '11 at 18:40
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That all depends what you mean by the number 1.

You see, mathematics is- associated by whatever means, and in whatever terms, the reader's philosophy dictates- a linguistic construction that reflects our intentional phenomena in regarding things.

But as such, just as one is duty bound in thought, regarding a single thing, only to have a phenomenal intention that incorporates that of 'a single thing', one must be duty bound in one's mathematics only to incorporate the mathematics of 'a single thing'. And the mathematics of single things, unconstrained in sub- and superstructure, is rich indeed- perhaps the whole of mathematics.

Is it still '1' if it is one hour on an analogue watch, where twelve is indistinguishable from nothing? Is it still '1' if it is each 6 hour leap of the same- where 1 and then 1 is indistinguishable from nothing? If it is one kahler manifold? The mereological 'Top' object? The number two? The category with one element? If it is simply 'a small amount', and more will make 'much'*?

If you answered 'yes' to all of the above, then perhaps. But the extent to which the existence of a thing 'proves' the existence of the number 1 (in any of its wide variety of senses) depends upon (can be 'proved by') a fact about the world depends in turn on the (as yet unresolved!**) question of how to conceive of the relationship of one's intentional states to the world.

* This is a reference to the Piraha tribe, see link posted in comments

** And I would not wait around for a consensus

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  • I did answer yes to all of those questions. As for my position, I think that the number 1 has the strongest possible form of existence. Although I do agree with your last paragraph and I think you summed up the essence of the question in that paragraph, I am trying to claim that our existence, including how our intentional states relate to the world, neccessarily depend on the existence of 1. The logic is this: "If I can only make claims which descriptively contain the concept blue, then blue must exist and blue's existence must be strictly stronger than the truth of my claim. " – REX Jul 25 '11 at 5:13
  • You have many (structural) characterization of the number 1. One can characterize what it means to be a recursive structure (that is, to be like the natural numbers), with simply an object N, a "starting point" (the zero) and a recursion function s. In this case, 1 is just s(0) and it is perfectly meaningful. That is, 1 is the generator of an additive monoid. Now, one could also characterize 1 entirely as the unit of the multiplicative integers, but the latter are much more difficult to characterize. – sure Sep 16 '15 at 8:09
  • @REX: to make sense of 1 as "there exists something" and call this something "1" is perfectly meaningful. Now, if you want to make sense of 2, you also need abilities to differentiate the things that are part of existence. To be is a whole: strictly speaking, what we notice of existence is everything interacting with us at once, that is, it isn't meaningful to differentiate one object with another and thus create the concept of "2", "3", "4", ... For that, one needs some intelligence and consciousness. – sure Sep 16 '15 at 8:13
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In a traditional formulation of Peano arithmetic (due to von Neumann, I believe), we start by assuming that nothing exists: i.e. the empty set exists. We can then define the successor of zero ("one") to be the set which contains the empty set and zero.

So if you mean "assume something exists" to be something like "assume the empty set exists and various set theory axioms are correct", your assumption will lead not just to the existence of "1" but of 2, 3, ... Conversely, if you deny the existence of some natural, then you must deny either that something exists or that these axioms are correct.

The point being: it depends on what you mean by "number." If you take this basis that I've described here, and you can say "assume X exists", then you can let X be your "zero" element, and inductively prove the existence of the naturals.

See Wikipedia for more.

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  • I think the important point is that particular numbers exist only in the context of particular theories such as Peano arithmetic or under certain axioms such as ZF set theory. But these are theories of natural or ordinal numbers. The OPs existential inference is less important than the mathematical context you have pointed out. – danportin Jul 26 '11 at 1:18
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The following may be an example of accepting a non-existent object to define an existent object. Perhaps it is an example of an unaccepted object used to understand an accepted solution. If not an answer, then a probing at the question.

In the 16th century, the formula for the solution to a cubic equation of the form $x^3 + px + q = 0$ was found (by an entanglement between three mathematicians: Gerolamo Cardano, Niccolo Fantana Tartaglia, and Lodovico Ferro: Cardano was Tartaglia's student and claimed Tartaglia's work to be his own which eventually led to dual and Ferrari was Cardano's student who eventually completed the solution to the problem of the cubic formula). For example, to solve the equation $x^3 - 3x = 0$ using the formula, one gets $x = \sqrt[3]{\sqrt{-1}} - \sqrt[3]{\sqrt{-1}}$.

Imaginary numbers were not widely accepted and used in mathematics until 200 years later, beginning with Euler. But, to solve this cubic equation and get that $x=0$ is a root, using the cubic formula, one had to accept the existence of $i = \sqrt{-1}$ (and therefore $\sqrt[3]{\sqrt{-1}}$) briefly, long enough to subtract it from itself.

i did not exist yet (in the minds of mathematicians), but it was used to find a real solution.

This view of $\i$ is due to Saul Stahl in his historical introduction to Algebra: Introductory Modern Algebra

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    However, I can make claims which do not involve a description of i, because 1 does not descriptely determine the theory of Universal Algebra. However, I cannot make any claims which do not contain the concept of 1. To me this is a different sort of claim. – REX Jul 25 '11 at 5:18
  • +1 for the mathmathical arguement for the existance of mathmatics. – Chad Jul 27 '11 at 21:01
  • Perhaps a more nuanced idea of non-existence should be in order. If something doesn't exist one cannot manipulate it. Rather, I think something here does exist but its nature is mysterious. A more contemporary example in physics is the conjectural M-Theory or the theory of motives in mathematics both of whose nature remains mysterious. – Mozibur Ullah May 7 '13 at 12:45
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To go in reverse order, you wonder if 'there exists' implies that there exists exactly one. No, the stipulative definition of the logical 'there exists' (used in mathematics or in ordinary discourse) says that there is at least one. In order to state that there exists exactly one, you need to specify additionally the 'exactly' part, or also say there is at most one.

As to your primary concern, I think there is overlapping but not identical use of the word 'exists' for numbers (like '1') and other things (e.g. friends, a verifiable theory, an afterlife, other minds, an apple in front us). Surely it is very immediate to say that 1 apple exists if it is sitting in front of us. But what does it really mean to say that '1' (oneness) exists? Does it exist in the same way as the single apple itself? What about negative 1? When the notation was introduced in Europe, there was quite a bit of controversy whether 1 was actually a number, and then later 0 itself and later negative numbers and complex numbers. But people got past all that (it is still a bit of a controversy whether one 'has' a negative number of apples).

Anyway, numbers are (ahem) one way of describing sets. Suppose the set 'exists' by one standard of existence. Then certainly the ability to describe sets using numbers comes with the ability to talk about sets at all which is a different thing than the circumstances of that particular set's existence. So in that sense, the existence of 1 (and other numbers) comes before your particular claim of existence.

Executive summary: Yes, '1' comes first (or rather together with the machinery of mathematical existence).

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Short answer: This reasoning is questionable and no, "there exists" does not mean there exists exactly one.

Longer Answer:

First, the reasoning is questionable because you're using 1 as a synonym for existence. However, how is this any different from using any other synonym for existence? Wouldn't the ontological status of 1 then be simply the same as the ontological status of any synonym for existence? What have you done to show 1's ontological status as a number? In all fairness, this has been a problem historically as the status of 1 as a number wasn't always accepted (ditto for 0 whose very invention was considered a major development in mathematics).

Second, "there exists" (the Existential Quantifier) is true if the property holds for at least one of the entities. So if the property holds for any number of entities (above 0), then it's true.

NOTE: This completely avoids the question of what existence means in this case. That's a can of worms I'm not about to open -- at least not in this response :)

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I have no background in analytical philosophy, so this may be why I find this mix of number theory and ontology confusing in an apple-oranges way.

In dismissing the ontological proof of God, Kant famously insisted that "existence is not a predicate." Once the thing is specified, we add no information by saying "and it exists." Of course, refinements in logic and set theory by Russell and others further clarified hidden assumptions of "existence."

It strikes me that your use of the number 1 by itself in a proposition is similar. Math doesn't get started until we have 0 and 1 or 2, a binomial system... apart from which "1" has no meaning and adds no information to "that thing" you are already specifying.

To infer its existence from a "thing" that exists sounds a bit like Anselm's ontological proof, to which an adaptation of Kant's rule "oneness alone is not a predicate" is at least commonsensical, if by no means definitive. Your 1 acquires meaning in relation to 2 or "other numbers." Whether or not numbers "exist" is a different question.

As I say, much of the analytical terminology above is unfamiliar to me, so apologies if I am way off base.

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    It's weird that you would invoke Kant here, because Kant also has a method called "transcendental arguments" which he uses to prove that time and space exist outside the subject. Arguably, you could say proving the existence of anything is also a transcendental argument for at least a weak concept of one. (though in Kant's case it may be a very strong one since one vs. many is one of the categories of understanding and a condition for knowing anything at all). – virmaior Sep 15 '15 at 17:19
  • Even while I say that, I'm no expert in math, so I'm not fully versed in some of the potential meanings hidden in the concept of the number one vis-a-vis contemporary math. – virmaior Sep 15 '15 at 17:19
  • I was not really invoking Kant in general. While Kant thought math was "synthetic a prior" I'm not sure what we would say about the "existence" of numbers. Kant's transcendental deductions secure possible knowledge and unify "apperception" and experience, but cannot be said to certify the "existence" of anything, hence his turn away from ontology to epistemology and the limits of metaphysics. My analogy is that just as "is-ness" was not meaningful for Kant, so "1-ness" in propositions, as above, is not by itself a valid argument for an existent "1." – Nelson Alexander Sep 15 '15 at 18:04
  • Kant definitely says that geometry is synthetic a priori, as is number; its Hilberts understanding of this that allows him to push away from the Frege-Russell logicist programme and formulate his own; I just saw a quote in a paper that quoted him on this, which surprised me since I had associated Kants approach to number with Brouwers intuitionism. – Mozibur Ullah Sep 15 '15 at 22:22
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It appears you're using the word one as a synonym of some thing exists.

Implicitly you are not asserting that it is everything - in Aristotles terminology, the All; whose concept includes existence.

Hence it must be individualisable - in the sense of being distinguishable from everything else; and this too is part of your concept of one.

In this bare sense of one, you are not asserting that it is unique - there may be others - in which case it would be 'one of'; but it also might be the case, that it is in fact unique - ie 'the only one'.

Mathematically, one is not bare in this sense; it has a range of meanings:

  • Operationally as the identity

  • Order-theoretic: it is the first

  • Change: the least difference

These properties define the Peano axioms; which is the formalised concept of the positive integers; arguably it is this context that we mostly think of when we think of the number one.

None of these properties are contained in the concept of the bare one. Hence, they are definitely not identical.

The standard symbol for 'there exists' is a reversed E; it may it may not exist uniquely - if it in fact does exists uniquely, then sometimes an exclamation mark follows the reversed E.

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Yes 1 exists. It is not a material thing however and instead immaterial. It is an "idea" of sorts, but one that exists nonetheless. Of course it all depends on what we mean by existence of course, but it would be best to apply it in it's broadest sense. In this broad sense clearly, 1 exists. An example of something that doesn't exist or rather and even more strongly, couldn't exist, is 'nothing'. Nothing cannot exist. Nothing, while like 1, being an idea, would still be something, namely an idea. Likewise if 'nothing' did exist somewhere etc. then it would be something and no longer 'nothing'. Some say the set of nothing or the null set would show nothing existing, however that is still a set and not a proper referant of nothing. Remember that nothing has no referant. Simply put you couldn't even think of nothing, you would fail to do so, and further you would fail to be DOING nothing, because you would be engaging in a thought process, namely thinking of nothing. All in all, nothing could not exist, BUT MOST ASSUREDLY NUMBERS EXIST. After all, life and everything for that matter is dictated by physics, and well there's math and numbers involved in that. Similarly math and numbers are real because they are not, as some like to believe, man made or arbitrary. We may NAME them differently or CALL them different names like one and uno or even the symbol 1 and I (roman numeral) however the idea is always the same, the meaning is always the same, 1. So, 2,3,4,5,6,7,8,9, and even 0 exist, and also don't think 0 is nothing, while it is the symbol for the idea of nothing, it itself is a thing and there is no true referant for nothing, because then it would fail to be nothing and be a thing.

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  • Hi @Ben. There is no need to use all-caps. If you want to emphasize a bit of text, you can put it between underscores or double asterisks. – user3164 May 7 '13 at 14:06

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