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Is it just two marks/numerals representing a singular number 1, or are they actually two instances of 1?

And what about in a set with repetition such as {1, 1, 2, 3}?

And if these are actually multiple instances of 1, how could that be, when in the domain of discourse concerning numbers, numbers are individual objects, which would mean they can't be instantiated?

J D
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csp
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  • I guess 1 + 1 cannot be 2, because how could the same thing be added to itself to give something else? Writing down the expression 1 + 1 is not even legit, unless those 2 1's are not the same thing, but if they are not the same thing, why are they written with the same symbol? – Frank May 27 '23 at 18:00
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    As far as the written string goes, they are two [occurrences](https://plato.stanford.edu/entries/types-tokens/#Occ) of "1." For e.g. {1, 1} there is a theory of [multisets](https://en.wikipedia.org/wiki/Multiset). – Kristian Berry May 27 '23 at 18:18
  • I know this exact same question has been asked within the last year, but I can't find it. The search feature on this site is pretty weak. – David Gudeman May 27 '23 at 19:53
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    Numbers are individual objects only under some sort of platonism. And in platonism they *can* be instantiated (or "partaken in", as Plato is translated to say), either by concrete things or by abstract particulars. Mathematicians usually refer to instances as "copies". 1 + 1 = 2 indicates that putting together two instances of 1 instantiates 2. – Conifold May 27 '23 at 21:01
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    @Conifold Do you have a reference for a mathematician who would say there are two "copies" of 1 in 1 + 1 = 2? The language of "copies" and "instances" doesn't seem to belong in ordinary mathematics (?) it seems to be a slippery slope towards giving numbers some kind of substance, which mathematicians probably are not very interested in. – Frank May 27 '23 at 22:36
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    @Frank Use of "copies" is common in "ordinary mathematics", see e.g. [Math SE](https://math.stackexchange.com/q/2838285/152568). So is naive platonism and its realist terminology more broadly. However, using such language does not commit one to metaphysics it comes from, and most mathematicians do not care to muse over established terminology. So the slope is not very slippery. – Conifold May 27 '23 at 23:06
  • @Conifold A _slippery slope_ for ... philosophers who want to adduce all sorts of metaphysics to the mathematics of mathematicians :-) – Frank May 27 '23 at 23:49
  • I am not a philosopher... Would this be determined by the answer to the mathematical question? Two references to the same instance would equal "1", if different instances would equal "2". ie "gin" + "gin" equals "gin", "gin" + "tonic" equals me looking for a lemon slice. – GeeTee May 28 '23 at 10:38
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    Mathematician here. What the heck do you mean by an "instance"? – Daron May 28 '23 at 12:40
  • @Daron I'm not entirely sure in this context but it seems to be something like a programming "instance" i.e. you create an object, that when instantiated becomes an instance that you can use without actually changing the code of the object itself. – stickynotememo May 29 '23 at 12:28
  • @stickynotememo If that is the case then my answer is still two. Two instances. – Daron May 30 '23 at 08:29

4 Answers4

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The TLDR

One can instantiate the numeral '1' in a sequence or multiset multiple times, in a set '1' only once, and the concept of 1 cannot be instantiated more than once ever, given how the concept 1 is grounded in Peano's Axioms, for instance.

Explanation

In philosophy, you are talking about the type-token distinction and the difference between the particular and the universal. At this point, there's terminology enough to make language articulate.

  • There is the concept of one which might be understood alternatively as "oneness" and exists compositionally in a proposition.

  • There is a the numeral '1' which is the grapheme that arbitrarily or amodally represents the concept.

  • There may be alternative ways of writing the one, such as in a different typeface, and such a '1' would be an allograph.

  • In other languages, there are different graphemes for one, such as in roman numerals (I) and Japanese (一). We often call these characters. Thus, '1', 'I', and '一' are distinct graphemes or symbols entirely but represent the same concept.

  • In philosophy, the terms reference and sense are often used to describe the difference between the label and what is labeled, resp. There are other philosophical term like symbol and referrent by Bolzano.

  • In linguistics and the philosophy of language, there is the distinction of syntax which is the form we represent the concept with, and the semantics which is the concept itself, and often the difference in use-mention entails using delimitters so that one and 'one' refer to the semantics and the syntax, resp.

  • In mathematics, a sequence or multiset means that multiple occurrence of the same numeral or grapheme represents different instances. Thus (1,2,1,3,1) might be read as (11,2,12,3,13). Each of these can be respresented as existentially distinct using what mathematicians call an index set, where the '1' in the index set is a counting number.

And if these are actually multiple instances of 1, how could that be, when in the domain of discourse concerning numbers, numbers are individual objects, which would mean they can't be instantiated?

So now, we have enough language to explain. If you have a sequence of numbers, say composed of the naturals (1,2,1,3,1), each numeral '1' is a symbol that refers to the sense of 'oneness' which mathematically is often defined these days by Peano's Axioms. The instances of numerals are not the same because the sequence is indexed as a mapping of (1,2,1,3,1)->(1,2,3,4,5). Since the first numeral one is the first element in the sequence, and the second is the third, and the last is the fifth, we say there are three instances of the same numeral that represent three different elements of the sequence and can index them using the naturals as (11,2,12,3,13). We use the Indian-Arabic numeral system, but in the same concept of "oneness" would be just as viably discussed in the Roman number system with the sequence (I,II,I,III,I), and we could even index them with Roman numerals (Ii,II,Iii,III,Iiii) and it would make no difference conceptually, because the grapheme is an arbitrary assignment of sorts (what one might call historical accident). We could use binary numbers too! Instantiation or reification in the philosophical sense means that one merely is producing multiple tokens of the same type, which is pretty close to how computer science uses the term. So, how we refer to them as duplicates, occurrences, instances, reifications, etc. recognizes the basic distinction between the syntax of the symbol and the semantics of the symbol. For instance, in a sequence, the various numerals '1' have different positions, and therefore different properties. Were they to be drawn by hand, there would be differences of each relative to each other, just as there are differences in concept between the ordinal '1' and the 'cardinal' one conceptually. Thus, a domain of discourse is nothing more than the context for talking about '1', 'I', one (as an entity), and "oneness" (as a property) in any discourse, and that domain determines precisely what instantiation means which depends on each on particular context.

J D
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  • Thanks, J D. So are you saying each 1 is a token numeral of the type numeral 1 which maps to the number 1 (whatever number 1 may be), and not that each token numeral 1 is mapping to different instances of the number 1? – csp May 27 '23 at 22:52
  • @csp Exactly. For clarity, I'd write. Each '1' is a token numeral of the type '1' which refers to the sense of one which can be understood conceptually by the dictionary definition, the definition given in PA, or even [von Neumann ordinals](https://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals). Furthermore, to a Platonist the tokens would be taken to be an imperfect copy of a Form that exists in the transcendental realm of Forms, but more contemporaneously, from a [nominalist(SEP)](https://plato.stanford.edu/entries/nominalism-mathematics/#ProUniSem) perspective... – J D May 28 '23 at 04:21
  • they would be fictions a la linguistic artifact that have practical value in discourse because they express similiarities between things that exist; the notion of type is a bit flexible because multiple tokens might be all of the same typeface, of several different typefaces, or several different graphemes. But this is the flexibility of our general intelligence, to shift the linguistic frame to suit our purposes. – J D May 28 '23 at 04:26
  • What was the purpose of bringing in the various different views on seperating reference and referrant in different fields? – Reine Abstraktion May 28 '23 at 08:45
  • @HopefulWhitepiller The concept of token and type is inextricably linked to representation. Take pennies. Each penny is a token and is representative of the type penny. There are no actual physical types pennies. The type penny is an idealization that is related to the collection of actual physical pennies each of which may vary in some small way (at the molecular level, for instance). But then we have the token 'penny' symbolically which refers to the penny, right? In German, one would use a different token: "Pfennig"... – J D May 28 '23 at 15:42
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    Consider the sentence "If one wants to discuss if one penny and one penny is two pennies in German, one would use 'Pfennig' rather 'penny' in constructing a sentence". What's important is keeping type and token, and symbol and referrent straight. There are three tokens of penny in that sentence, right? But the token penny in "'penny'" refers to the tokens "penny" and "penny" earlier in the sentence according to use-mention. So, the example has three sequences of (p,e,n,n,y), however one of them is nested like (',p,e,n,n,y,') to show it is an abstract symbol that refers to a concrete referrent. – J D May 28 '23 at 15:57
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    Since use-mention and type-token are often discussed about the same material, it's important not to confuse the two. Use-mention is about reference and sense or symbol and referrent, and type-token is about particulars and a universal or in mathematics, elements and sets. Combining use-mention and type-token conceptually can get confusing, so it's important to have the language to disambiguate. :D – J D May 28 '23 at 16:01
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Two Instances

This is easy. Zoom in for a closer look.

enter image description here

Fig. A

In the figure I have zoomed in and circled and labelled both instances of 1 in the expression. I think I got all of them. Perhaps more become visible at higher magnification.

Daron
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    Don't know why this got so many upvotes. OP's question clearly implicates the use/mention distinction, and it's clearly not possible to "circle" the number 1, it's possible only to circle *mentions* of it. – Acccumulation May 29 '23 at 02:27
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    @Acccumulation I circled "the" instances of the number. Not the *mention*s. – Daron May 29 '23 at 07:38
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Here's a computer scientist's view:

"1+1" in this context is a mathematical expression, consisting of the following terms:

  • "1": terminal symbol, can be directly evaluated to the value 1
  • "+": binary function, processing the previous and the following term under the commonly known rules of addition
  • "1": (again) terminal symbol, can be directly evaluated to the value 1

finally, applying addition to the values 1 and 1 evaluates to the value 2, leaving "2" as the final result of the given expression.

So, how many instances of 1 are present depends on whether you are counting values or references. Two symbols represent the value 1, but the value 1 itself is unique and, since the expression does evaluate to 2, I would argue that the value 1 is not present in the expression at all, because in order to consider the value of an expression you first need to evaluate it.

Florian Herzog
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  • Thanks, Florian. I think this is similar to what J D said, but cool to see it in a different perspective. – csp May 28 '23 at 15:28
  • It's maybe worth elaborating on this as, are there 1 or 2 instances of x in the expression x + x? What if x is let-bound to a more complex arithmetic expression requiring further computation, some of which might have side-effects? You can view the expression as a tree or, if you've got an optimizing compiler capable of common sub-expression elimination, as a directed acyclic graph. The answer is different in each case. – Daniel Shapero May 29 '23 at 18:42
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If there were only one instance of one in one plus one then how would one distinguish that from the one instance of one that appears in one by itself?

Marco Ocram
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