# In what sense are mathematical relations necessary?

I just stumbled upon the following statement:

(1) The number of discovered chemical elements is 118. Take the sentence "The number of chemical elements is necessarily greater than 100". Again, there are two interpretations as per the de dicto / de re distinction. According to the de dicto interpretation, even if the inner workings of the atom could differ, there could not be fewer than 100 elements. The second interpretation, de re, is that things could not have gone differently with the number 118 turning out to be fewer than 100. Intuitively, this claim is true. Of all the ways the world could have turned out, presumably there are no possibilities wherein 118 is fewer than 100. That 118 is greater than 100 is a necessary fact. [Wikipedia]

In one of the answers to a related SE question, somebody said:

(2) There exist true mathematical statements. They are true in all possible worlds where our logic is valid, which means necessarily true.

`100 < 118` is obviously a true mathematical statement in our world, given that `<` denotes a standard order relation on natural numbers. However, I can easily imagine a possible world where our logic is valid, still, standard `<` is defined to mean, e.g.:

``````... < 99 < 118 < 101 < ... < 117 < 100 < 119 < ...
``````

Is there any problem with that? The condition (2) is satisfied, so can we conclude that the statement `100 < 118` is actually contingent? Is (1) incorrect, or is (2) incorrect, or am I missing something?

EDIT: What is it exactly that makes de re reading true, and de dicto reading false?

• Being necessarily true assumes that we agree on the meaning of words. If you change the meaning and redefine 100 to signify 118, it is necessarily true that it is greater than 117 for all those that agree to use that particular meaning of 100. – user3017 Mar 30 '17 at 12:44
• @PédeLeão But is it really what I did - just renamed 100 and 118? If so, couldn't we apply the same to the de dicto interpretation of (1)? What if I define `<` as: `118 < 1 < 2 < ... < 100 < ... < 117 < 119 < ...`? This is not renaming. – machaerus Mar 30 '17 at 13:09
• "However, I can easily imagine a possible world where our logic is valid, still, standard < is defined to mean..." Of course, but that is not "standard" < any more. – Mauro ALLEGRANZA Mar 30 '17 at 13:21
• The natural numbers, at least, can be defined with purely logical notation. To say that there is one x characterized by some property P is to say: Ǝx[Px & ∀y[Py → x=y]]. That means that its meaning can be understood in terms of the logical concepts of existence, identity, quantification, etc. The basic relations of arithmetic can also be defined with logic, so its necessity is rooted in the necessity of logic, which, in turn, is analytic in nature. That basically means that it's the study of saying the same thing in different ways. – user3017 Mar 30 '17 at 16:37
• @PédeLeão It is generally believed that the principle of induction is an integral part of our conception of arithmetic, and that this principle is not reducible to logic (there are doubts that identity is pure logic too, especially as used in mathematics for abstracted equivalence). The Frege-Russell idea of reducing mathematics to logic did not work out, see logicism . – Conifold Mar 30 '17 at 22:33

There are two separate questions here, but I will answer them in turn.

1. Can we conclude that the statement 100 < 118 is actually contingent?

No. There is some confusion here about what is meant by mathematical statements being necessary. What is necessary about "100 < 118" is the relation between the two numbers, not how we symbolize this relation. Of course, other symbolizations are possible, but that does not change anything about the objects or the relations between them.

Here's another way to look at this issue, and to show that the question of necessity is independent from the possibility of change in meaning. We can distinguish two senses of necessity:

(a) "100 < 118" necessarily expresses a true proposition.

(b) "100 < 118" expresses a necessarily true proposition.

This is a subtle but very important difference. The first statement (a) is false, as your argument shows, since the symbols could have had different meanings. But (b) might still be true, and that is what is meant when it is said that "100 < 118" is necessary. The truth of (b) is independent of whether or not the symbols could have had different meanings; it only says that, given its current meaning, the relation expressed by the sentence necessarily holds.

1. What is it exactly that makes de re reading true, and de dicto reading false?

The simple answer is: scope. Take the example you cited: "The number of chemical elements is necessarily greater than 100". The two interpretations, de re and de dicto, are merely the following two ways of understanding the statement:

De re: There is an x such that x is the number of chemical elements and x is necessarily greater than 100. (True)

De dicto: Necessarily, there is an x such that x is the number of chemical elements and x is greater than 100. (False)

The crucial difference is where "necessarily" is placed. In the de re instance, necessity is attributed to the object (the number), while in the de dicto case it is attributed to the whole statement. This is what accounts for the difference in truth value.

• Thanks for clarifying this. I still have problem with the first part. Certainly, we can point to a pile of apples and say: "That amount of apples is greater than that amount of apples if we measure it in that way". But objects and relations involved in "100 < 118" are mathematical entities, not physical objects, so I feel there's not much of an objective reality behind them we could point to. This is what confuses me. – machaerus Mar 30 '17 at 15:08
• @machaerus You have to distinguish the string "100", which is a construct of language, from the number 100, which is taken to be, by most philosophers, an abstract object, whose existence is independent of language. Of course, whether this is true is a very big question in philosophy, and there's a lot to be said about it, but it is too big to be addressed here. I'll add a bit to my answer to try to clarify a little more. – Eliran Mar 30 '17 at 15:41
• @machaerus Also, you might find this question and its answers helpful (about the existence of non-physical objects): philosophy.stackexchange.com/questions/34031/…. – Eliran Mar 30 '17 at 15:59
• "What is necessary about "100 < 118" is the relation between the two numbers, not how we symbolize this relation." no. there are no numbers here, only symbols. the std interpretation of mathematical symbols is only one of infinitely many. put different, there is nothing necessary about interpreting the symbol "118" as any particular number. mathematical statements are never logically (i e necessarily) true. – user20153 Mar 30 '17 at 19:33

In philosophy, a lot of original thoughts began with a feeling of itch followed by frantic scratches on the outside of the boot.

I guess your question involves the following questions:

1. What is mathematics?
2. What do we mean by necessary?
3. What is the nature of magnitude, quantity, measurement and counting?

The following answer is from my understanding of the Russellian school:

1. Mathematics is the study of saying the same thing in different ways.

People, after they finished defining number and <, are delighted to find out that 100 < 118, but the fact is, 100 < 118 is already contained in the definition of number and <.

2. When a proposition necessarily follows its premises, it does so in virtue of logic, not in virtue of its relation with the objective world.

100 < 118 is necessary due the definition of number and <, not to its relation with the physical world.

3. I myself is currently trying to figure it out.

• Well Russell was a kind of guy who was delighted to find out that 1+1=2. :) After all we don't need mathematics to tell us that 100 < 118, quite the opposite - I'm pretty sure we had to know it before we put it into mathematics. So in general I think I agree. – machaerus Mar 30 '17 at 17:12