Gottlob Frege:

The Foundations of Arithmetic

How can we define the number 1?

Background

A few years after writing Concept Script, Frege wrote The Foundations of Arithmetic. In the work, Frege argues against many of the conceptions of arithmetic and numbers which were popular at the time, most notably those of John Stuart Mill and Immanuel Kant.

He argues that the foundations of arithmetic are not to be found in our psychology, but instead in logic.

He develops an ingenious approach to arithmetic which was very influential on later philosophers like Bertrand Russell and Ludwig Wittgenstein.

Frege's project

Frege begins the work with his central question: how are we to define the number 1? Normally, he says, we will try to explain the meaning of the number 1 by pointing to an example of one thing. But Frege is not convinced by this approach...

When we ask someone what the number one is, or what the symbol 1 means, we get as a rule the answer “why, a thing”. And if we go on to point out that the proposition

“The number one is a thing”

Is not a definition, because it has the definite article on one side and the indefinite on the other, or that it only assigns the number one to the class of things, without stating which thing it is, then we shall very likely be invited to select something for ourselves —anything we please—to call one. Yet if everyone had the right to understand by this name whatever he pleased, then the same proposition about one would mean different things for different people, —such propositions would have no common content.

The Foundations of Arithmetic, Introduction.

Kant's view of numbers

Kant believed that propositions could be either be:

Analytic: where the subject and the predicate were basically describing the same thing (e.g. a triangle is a shape with three sides), or...
Synthetic; where they weren't (e.g. the mug is on the table).

He also believed they could be:

a priori: where the truth of the proposition didn't need to be checked through observation or experience, or... (e.g. a triangle is a shape with three sides), or...
a posteriori: where it did (e.g. the mug is on the table).

Kant argued that there was a special class of propositions called synthetic a priori propositions which we somehow programmed into us, almost as part of the operating system of our mind. These ideas were, he believed prerequisites for making sense of our experience. He believed propositions about causation and morality fell into this group. He also believed that the propositions of arithmetic were synthetic a priori. 2+5=7 was not, Kant thought, analytically true, even though we didn't need evidence to know that it was true. He did not think that 2+5 was describing the same thing as 7.

Frege disagreed with this view entirely. He wanted to show that propositions of arithmetic were like a triangle is a shape with three sides, i.e. that they were analytic a priori. Frege wanted to show that 2+5 was describing the same thing as 7.

A number as an object

If the propositions of arithmetic were to be shown to be analytic, the first thing Frege had to do was show that a number is an object in the same way that a triangle is an object.

Let's take the following proposition as our model of what Frege was trying to show:

A triangle is a shape with three sides.

Here it's quite clear that a triangle is entirely synonymous with a shape with three sides. We can rewrite it as:

A triangle = a shape with three sides

(To demonstrate this, he uses the example of 'Jupiter has four moons' which is a bit out of date. Currently, Jupiter is thought to have 95 moons, but let's imagine that we don't know about the other 91.)

Frege shows how we can rewrite the sentence so that it has a similar form to the proposition above:

Jupiter has four moons
The number of Jupiter's moons is four
The number of Jupiter's moons = 4

…the proposition “Jupiter has four moons” can be converted into “the number of Jupiter’s moons is four”. Here the word “is” should not be taken as a mere copula, as in the proposition “ the sky is blue”. This is shown by the fact that we can say: “the number of Jupiter’s moons is the number four, or 4”. Here “is” has the sense of “is identical with” or “is the same as”. So that what we have is an identity, stating that the expression “the number of Jupiter’s moons” signifies the same object as the word “four”.

The Foundations of Arithmetic, Section 57

One to one correspondence

So, let's imagine that there are two groups: group A is 'the number of Jupiter's moons' and group B is 'the number of sides of a square'. According to Frege, we can define a number in terms of the one-to-one correspondence between these two groups.

So all the sets of the same size are the same number.

This gives rise to a central plank of Frege's system. The idea that

there can be a set of things which are F which can contain the same number as the set of things which are G.

The extension of the concept “equal to the concept F” is identical with the extension of the concept “equal to the concept G” is true if and only if the proposition “The same number belongs to the concept F as to the concept G” is also true.

The Foundations of Arithmetic, Section 69

In search of an analytic a priori definition

However, 'the number of Jupiter's moons = 4' is obviously not an analytic a priori proposition. We only know the number of Jupiter's moons through observation (and we were horribly wrong at the time that Frege was writing).

So Frege needs to somehow replace this observation with something a priori, something purely logical. He needs to define 1 in a way that doesn't require observation. His solution was ingenious.

He doesn't begin with 1, instead, he begins with 0. He defines 0 as being the number of things that don't equal themselves.

Obviously nothing is not identical with itself, hence the group of things that don't equal themselves is completely empty.

Since nothing falls under the concept “not identical with itself”, I define nought as follows:

0 is the Number which belongs to the concept “not identical with itself”

The Foundations of Arithmetic, Section 74

...and how does he get from this definition of 0 to a definition of 1?

Well the empty set (the set of things that don't equal themselves) is the same whether it's a group of no cars or a group of no teachers or a group of no tables. a group of nothing is always a group of nothing - so there is only 1 group of nothing. Thus, Frege can define 1 as the number of empty sets.

1 is the Number which belongs to the concept “identical with 0”

The Foundations of Arithmetic, Section 77

A problem

Frege's work wasn't particularly famous in his lifetime, one of the few people who was actually a fan was a British philosopher called Bertrand Russell. However, Russell, after reading Frege's Begriffsschrift noticed a flaw in the set theory underpinning Frege's approach.

The set theory described by Frege relied on an assumption that for any description, there will be a set of things that fall under it with a defined number.

This was the assumption that enabled him to say that:

there can be a set of things which are F which can contain the same number as the set of things which are G.

However, Russell pointed out that this basic assumption might not be quite right. Can a group, for example, contain itself? (A silly example: the Group of things beginning with G, contains the Group of things beginning with G.)

Imagine there is a group of things (let's call it group R) that only contains those groups which don't contain themselves. Does R contain itself? If not, then R must be in the group, and if so, then it must not –either way results in a contradiction.

A colloquial version of this paradox was once told to Russell:

The barber is the "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself?

There is just one point where I have encountered a difficulty. ... Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself? From each answer its opposite follows. Therefore we must conclude that w is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection does not form a totality.

Bertrand Russell's letter to Frege

Frege read Russell's letter just after he had written a work entitled The Basic Laws of Arithmetic. This book contained basic law V:

What holds of all objects, also holds of any.

The Basic Laws of Arithmetic, Section 20

It was pretty clear that Russell's paradox undermined this, as well as the assumption above that for any description, there will be a set of things that fall under it with a defined number just didn't really hold.

Frege was clearly quite shaken by Russell's paradox, as he wrote in the appendix to the second volume of The Basic Laws of Arithmetic:

Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr Bertrand Russell, just when the printing of this volume was nearing its completion. It is a matter of my Axiom (V). I have never disguised from myself its lack of the self-evidence that belongs to the other axioms and that must properly be demanded of a logical law. … I should gladly have dispensed with this foundation if I had known of any substitute for it.

Appendix, The Basic Laws of Arithmetic, volume II

Tasks

What was Frege trying to do?
What did Kant think about numbers and why did Frege disagree?
Why did Frege think we could think of numbers as objects?
How did Frege define 1 in the end?
What was the problem that Russell found with Frege's approach?

#logic

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