6 min read

The Philosophy of Arithmetic - 1

I wanted to learn more of Husserl's story about how we deal with numbers bigger than twelve, where we can't perceive the exact number in one go.

I'm interested not because of this particular question but because it stands in for his whole story on 'authentic' and 'inauthentic' representations, which is relevant to cognitive decoupling and Derrida on indication and expression and probably some other stuff I want to know about. So I've ended up going back to the source, The Philosophy of Arithmetic. This is from fairly early on in Husserl's career, after he got his PhD in mathematics and started working seriously on psychological foundations of maths, but about ten years before Logical Investigations which is what Derrida is mainly talking about in Voice and Phenomenon.

Husserl isn't exactly a riveting read but The Philosophy of Arithmetic is better than I expected, because he uses a fair number of examples and the discussion is fairly concretely grounded - what happens when we see a bunch of things? what happens when we try to count them? The tone of it is a bit more 'psychology' and a bit less 'philosophy' than Logical Investigations, which suits me.

The first part of the book is about what Husserl calls 'authentic representations', things we can intuitively perceive directly. I've skipped straight to the beginning of the second part, which is where his story on how we build inauthentic representations out of authentic ones starts. The rest of this post is just me pasting in quotes, maybe with some comments.

... we shall, in the further course of our investigations, supply yet more direct positive proofs that these "new turns," these diverse "forms in the composition of units," are nothing more than turns and forms of the symbolism, grounded upon the fact that all operating which reaches beyond the very first numbers is only a symbolic operating with symbolic representations.
If we had authentic [eigentliche] representations of all numbers, as we do of the first ones in the series, then there would be no arithmetic, for it would then be completely superfluous. The most complicated of relations between numbers, which now are discovered only laboriously, through intricate calculations, would then along with the number representations be simultaneously present to us with that same intuitive Evidence as we have, say, with propositions such as 2 + 3 = 5. To those who know what "2," "3," "5," and the signs "+" and "=" signify, this proposition is immediately clear and Evident. But in fact we are extremely limited 5 in our representational capacities. That some sort of limits are imposed upon us here lies in the finitude of human nature... Nevertheless, finite beings would be conceivable who could achieve actual representation of the millions and trillions - indeed, even the light years of the astronomers. Such a case would suffice to deprive the development of arithmetic of any practical occasion. Indeed, the whole of arithmetic is, as we shall see, nothing other than a sum of artificial devices for overcoming the essential imperfections of our intellect here touched upon.
But how can one speak of concepts which one does not genuinely [eigentlich] have? And how is it not absurd that upon such concepts the most secure of all sciences, arithmetic, should be grounded? The answer is: Even if we do not have the concept 5 given in the authentic [eigentlicher] manner, we still do have it given - in the symbolic manner.

This is the point he mentions Brentano in a footnote - 'In his university lectures Franz Brentano always placed the greatest of emphasis upon the distinction between "authentic" and "inauthentic" or "symbolic" representations'.

We have, for example, an authentic representation of the outer appearance of a house when we actually look at the house; and we have a symbolic representation when someone gives us the indirect characterization: the comer house on such and such side of such and such street
A determinate species of red is authentically represented when we find it as an abstract Moment in a perception. It is inauthentically represented through the symbolic determination: that color which corresponds to so-and-so many billion vibrations of aether per second. If we associate the name "triangle" with the concept of a closed figure bounded by three straight lines, then any other determination which belongs in univocal exclusiveness to triangles can stand in as an adequate sign for the authentic concept - e.g., that figure the sum of whose angles equals two right angles.
External signs can also serve in symbolization. Thus, by "C3" the non-musician will merely represent to himself the indirect characterization: that tone which musicians indicate by means of the sign "C3." Psychologically considered, external signs mediate every time language comes into play. But so far as logic is concerned, such signs come into consideration only in cases where the concept of that which is to be designated as such by an external sign belongs to the essential content of the symbolic representation.

Representing multiples

There's lots of stuff on what we're doing when we see multiple things. We can intuitively grasp that 'here is a bunch of the same sort of things', even if we can't tell how many there are. I got lost in the weeds quickly but here are a few quotes.

Our intention is directed to such union, but we lack the corresponding mental capacities to completely satisfy it with larger groups. Indeed, the successive grasping of the terms in the group one by one is still possible, but no longer their comprehensive collection. And, insofar as we in such cases still speak of a group or multiplicity, this obviously can only be in the symbolic sense.
But we also commonly stay with symbolic representations even where an authentic group representation could still be formed. And where that is not possible, we still do less in the way of authentic activity than we would be capable of.
We step into a large room full of people. One glance suffices and we judge: A group of people. We look up into the starry heavens, and in one glance judge: Many stars. The same holds true for groups of wholly unrecognizable objects. How are such judgments possible?
That "one glance" of which we spoke above is not to be taken in an entirely rigorous sense. The apprehension is not exactly instantaneous, and we sometimes note how the moving eye on its own picks out this or that individual object or a small cluster here and there. Instead of carrying through the entire process of collection, we thus content ourselves with the mere rudiments of one. We seize upon whichever of the individual objects directly impose themselves, conjoin them, but break off again immediately by forming the surrogate representation: a total collection of objects which would be produced by fully carrying out the process just begun.

Hm, so it turns out that this multiples thing is a whole chapter and I'm already getting lost. For this initial skim I'm going to skip the rest, and also the next chapter, and start on Chapter 13, 'The logical sources of arithmetic', which seems closest to the counting-past-twelve thing. I'll probably need to go back later.

The logical sources of arithmetic

Starts by pointing out that arithmetic is the science of the relationship between numbers, more than just the science of numbers. Also that arithmetic and calculational technique are often identified.

Let us now consider the concept of calculational technique [Rechenkunst]. It is given to us when we possess that of calculation. The concept of calculation, however, admits of various wider and narrower significations. By calculation in the broadest sense one can understand any mode of derivation of numbers sought starting from numbers given. Accordingly, we certainly would already have to call the unification of the numbers 2 and 3 to form 5, on the basis of the authentic representation of the concepts themselves, a calculation. Likewise for the construction of the concepts of systematic numbers, whether we adopt the path of concept formation or that of mechanical-exterior sign formation.

Hm that big piece of new jargon, 'mechanical-exterior sign formation', is making me think I skipped too far forward. And... yeah... scrolling back the previous chapter looks more relevant. Sorry to anyone who is actually trying to follow this for some reason but this notebook is for real-time discovery.

The symbolic representations of numbers

Yeah, this is the stuff, here's the twelve thing:

The symbolic representations of groups form the foundation for the symbolic representations of numbers. Had we only the authentic representations of groups, then the number series would at best end with twelve, and we would not even have the concept of a continuation beyond that. Along with the obvious lack of restriction on the symbolic expansion of groups, the same is also given for numbers, as we will soon see.

I'm getting tired and it's getting towards the end of Bristol library study hall so will finish here for now. At least I'm feeling better oriented and know where to start next time.