6 min read

The Philosophy of Arithmetic - 2

OK, back to it. I want to understand Husserl's story on how we build 'inauthentic representations' of numbers on top of 'authentic representations'. Last time I got as far as finding the right part of the book, Chapter XII 'The Symbolic Representations of Numbers', now let's try actually reading it.

The numbers are the distinct species of the general concept of multiplicity. To each concrete multiplicity, whether it be authentically or symbolically represented, there corresponds a determinate multiplicity of units: a number... In the symbolic sense we thus can say of any arbitrary group that a determinate number accrues to it even before we have formed that number itself; indeed, even when we are not in position to undertake the actual [wirklichen] formation of it. Likewise we may with good reason state that two arbitrary groups must be of the same or of a different number, whether we can conceptualize the number or not.

Oh great, I've just noticed that Husserl is about to start using 'Ideally' and 'Idealization' with a capital 'I'. Like this:

We are even justified in judging that the domain of number encompasses an unbounded manifold of species. In fact, if we start with any arbitrary symbolic group representation, then we possess (at least Ideally) the capability of expanding it without limits by continually adding new and ever new members. If we can do nothing else, we can at least think of the members proper to the group as mirrored in a constant reiteration, and accordingly form the concept of the progressive expansion of the group by means of the members of its reflections. Certainly this symbolic concept formation involves a strong Idealization of our powers of representation.

This suggests that a worrying amount of metaphysics is coming up :(

I'll keep plugging on though, this next bit seems OK otherwise:

Indeed we cannot, in fact, form the required reiterations in infinitum and arrange them in sequence. We lack time and strength for the ever-renewed mental activity required, as well as symbols for keeping the formations of those reiterations distinct. However, we can, by way of Idealization, disregard these limitations on our abilities and conceive the concepts, which are symbolic also in this respect. If now the given group is symbolically expanded through such means, then there belongs - again, in symbolic representation - to each level a determinate number, different for each new level.

Next section is on 'non-systematic symbolizations of numbers'. Starts with authentic representation of small numbers, e.g. 4+3, where we can authentically understand both each number and their sum. Then generalise to sums where we can authentically represent the individual numbers, but not the sum:

Assuming that ten was the last authentically representable number, then there are various possibilities for denumerating those groups that are not exhausted by the numbers up to ten. Any arbitrary decomposition of the group - or any decomposition of it that is of itself suggested by the character of the group intuition - into authentically enumerable partial groups leads to the symbolic construction of the concept of a number which is additively composed from the authentically representable numbers of the partial groups. What we thus grasp in concrete cases we can generalize, and there result symbolic number formations such as "10 + 5," "9 + 6 + 8," "7 + 10 + 5," and the like.

Next, the importance of having names for numbers, so everything doesn't get muddled together in a messy sludge:

In these formations an important function is fulfilled by the number names or number signs. In spite of the articulations, we can no longer hold such large groups of units clearly distinct in unitary representation. The composition of the signs is our crutch. As we reflect step by step upon their signification, the individual numbers in the sum enter our consciousness in the form of a determinate succession. Even if, as the new number turns up, the previous one blurs into obscurity - and, accordingly, the actually [wirklich] intended sum-representation cannot be realized - we still have the sensible composition of the names (or written symbols) as the fixed framework within which the succession of the conceptual elements in the sum, mediated through the symbolization, can always be generated in the same determinate manner.

This is nice actually. I keep poking fun at the number twelve thing, and the weird idealist philosophy, but I do actually like Husserl and I really think he was doing important work here. I'm interested in how symbolisation works as a technology to stabilise meaning and it's good to understand the history.

OK, so by 'non-systematic' he seems to be talking about this kind of naive process of globbing together small, authentically representable numbers and giving the resulting sum a name. Say 10+5 is, I don't know, 'glub', 3 + 8 + 4 = 'blargh', etc. Obviously this has limitations, like a lot of repetition:

One immediately notices that this procedure, which first offers itself for singling out determinate, symbolic number forms, is still highly incomplete. If we split up larger groups into partial groups, each of which does not exceed the number ten, we soon arrive at so many repetitions of the same partial numbers that we are hardly better off with the distinguishability of the emergent symbolic formations than with the corresponding unarticulated sums of units.

This can be tamed somewhat by adding only two numbers at a time. But that restricts us to 10+10 and below, if you want the numbers in the sum to be authentic (10 is the cutoff in Husserl's example here, not 12). So to go further, you need to construct 'towers' of sums, where only the initial sum is authentic:

Proceeding in this way, one could even restrict oneself to sum formations of two terms. If, for example, one has introduced the symbolic formation "p = 10 + 5," one can then construct, say, "p + 8 = p'," and then again "p' + 10 = p"," and so forth, whereby each later construct has the whole sequence of earlier ones for its basis.

We're still going to run into problems quickly though... a tower of like ten of these is going to completely exhaust our memory. Also comparison is going to be a mess:

One and the same group admits of various articulations, each of which will lead to a new symbolic number form; while the identity of the actual number corresponding to them all is guaranteed through the identity of the group in question. But one certainly would never suspect this from the diverse forms (e.g., 10 + 5, 9 + 6, 8 + 2 + 5, and so on). Accordingly, such systemless sum formations are totally useless for the purposes of number comparison. If we use the form p + q for one group, and the form p1 + q1 for a second, then merely by looking at them we cannot yet decide whether or not the same number corresponds to the two groups, or to which the greater and to which the smaller number corresponds.

Natural numbers

So we're going to need a consistent, systematic way of constructing numbers. Do this by adding 1 each time:

All of these requirements can be most simply satisfied, as is already done for the most part, by the procedure of successive number formation through the addition, in each case, of one unit to the number already formed. If we think of the authentically given numbers as so arranged that each arises out of the preceding one by increase of one unit, then we obtain the sequence:

1; 2 = 1 + 1; 3 = 2 + 1; 4 = 3 + 1; . . . ; 10 = 9 + 1.

That this sequence (taking ten to be the final number authentically representable) can be symbolically continued is clear. Certainly we can immediately form the inauthentic representation of a new number which issues from ten as ten does from nine: viz., through addition of one unit. If we call the number which is symbolized by the sum 10 + 1 "eleven" (11), then that number is given and defined by the equation 11 = 10 + 1. Likewise we can further define: 12 = 11 + 1, 13 = 12 + 1, 14 = 13 + 1, and so on.

Still have to solve the problem of naming though:

Each new step of symbolic number formation requires a new step in naming. If we were to choose ever new names (and that surely is unavoidable), then this would entail burdens for our memory which, already with numbers we currently are accustomed to regarding as moderate in size, would be unmanageable. And it would never be able to bear the load if all the new names necessarily had to be independent of each other: if we could not succeed in symbolizing all numbers of the sequence by means of a limited - and not too large - number of basic signs, following a uniform, easily understandable and clearly surveyable principle.

Dumbest possible idea of counting 'one, one one, one one one...' is also going to get cumbersome very quickly, and also practically indistinguishable - can you notice the difference between 'one' repeated 18 times and 'one' repeated 19 times?

How are we to find a transparent, simple principle which permits us to 25 construct, from a few basic signs, a symbol system that confers on each determinate number a convenient and easily distinguishable sign, and simultaneously distinctly marks its systematic position in the number sequence?

Skimming forward the next bit looks technical so I'll leave this question for next time.

Stray thought: could be interesting to compare this to Orality and Literacy, which is about the same process of symbolic technologies for language. Ong's book is much more historically grounded, ofc, this is more of a 'armchair reconstruction' type thing. I could read an actual history of number systems or something instead, but then I'd be missing the 'phenomenological side' which explains how the systematic concepts are grounded in intuitive ones. Ong sort of has both, which I like.

Another maybe-interesting comparison: Hasok Chang's Inventing Temperature. It's a while since I read that, but it talks about the history of building thermodynamics on top of intuitive ideas of heat and cold.