Logical and Spiritual REFLECTIONS
Book 2.A Short Critique of Kant’s Unreason
Chapter 7.How numbers arise
If we pay attention to theratiocinative acts at the foundations of mathematics, we notice the following intentions or mental movements:
First, there is the mentalisolationof something from its immediate experiential context, intending “thisand notthe rest of it”. A part of present experience is focused on, mentally delimited and considered virtually apart from the other parts of present experience. This makes possible the formation of the mathematical concept of ‘one’ (symbolized by a ‘1’), which is the elementary unit at the basis of all subsequent rational activities of computation or calculation.
Second, there is the mentalconjunctionof a selection of two (or more) such units, intending “thisandthat (or those)”. The units concerned may all be present in current experience, or some or all of them may have to be brought to mind by memory. This is the basis of the mathematical concept of ‘addition’ (symbolized by a ‘+’), which gives rise to compounds of units, i.e. natural numbers greater than ‘one’ (viz. ‘two’, ‘three’, etc.).
Third, there is the mentalidentificationof two (or more) such selected units, intending or declaring them to be ‘effectively the same’, ‘numerically equivalent’, ‘quantitatively equal’ (this being symbolized by the ‘=’ sign). Equivalence or equality between two items means that either item can in practice besubstitutedfor the other with regard to numerical value or quantity, even though symbolically they may be differently constituted (e.g. as two ones and one two).
Thus, first comes the idea of “1”.
From this, we build up the series of natural numbers, two, three, etc., by a succession of additions and equations, each of which relies on the preceding, ad infinitum: 1=1; 2=1+1; 3=2+1; 4=3+1; … etc. This gradual and infinite construction is enshrined and recalled in the process ofcounting: 1, (+1=) 2, (+1=) 3, (…) 4, (…) 5, 6, 7, … etc.
These definitions allow us to work out simple arithmetical inferences or proofs, by way of various appropriate substitutions. For example: 2+2 = 4 is derived from 2+2 = 2+(1+1) = (2+1)+1 = 3+1 = 4. The brackets ‘( )’are here used to signify our changing mental focus on the numbers involved.
We are now able to invent a fourth useful mathematical operation, viz.subtraction. This formalizes the idea of mentalexclusionof some unit(s) from a set of units under consideration. Thus, for example: having two things, removing one, leaves one; this is symbolized (using the ‘minus’ sign, written ‘–’) as 2–1=1. Similarly: 3–1=2; 4–1=3; and so forth (compare to the series of additions and their derivatives).
A special application of this idea of mental removal yields our concept of ‘zero’ (symbol ‘0’). “Given just one thing under consideration, if we remove this one thing, what are we left with?” The answer we give to this question is “zero”, which construct we interpret as thenegationof all other numerical concepts. That is, we intend by this ‘number’ an absence of any unit or collection of units (and later even of fractions of units, etc.); this is how we define zero.
In the same manner, the operations of ‘multiplication’ and then ‘division’ can be introduced, and mathematics as we know it can be gradually built up.
All this is simple and straightforward enough, and generally well known. Granting this obvious account, our next question has to be: “is mathematics, then, something ‘subjective’ or ‘objective’?” That is, what is theontological statusof numbers?
Historically, mathematics no doubt stems from the material experiences and needs of humans. We can well imagine how necessary economic activities likeisolatinga cow from a herd, orgatheringcattle in a pen, orexchanginga cow for some sheep, would in time (as life increased in complexity) give rise to the more abstract ideas of mathematics described above.
Raw experience is evidently non-numerical. It is just a whole, without parts. Nature itself is continuous. It is we, who experience it, who mentally cut the whole into parts, which we mentally regroup in various ways. We do not, however, engage in such acts randomly and arbitrarily, but (more or less) intelligently and rationally.
Without humans (or beings with similar cognitive and volitional powers),there would be no numbers, since the unity of all things would never be put in question. It takes humans to discriminate between various aspects of the variegated whole, and thus distinguish ones within a background of many, and so forth.
But this construction of number is neither entirely objective nor entirely subjective. That is, it is not given in experience in the way of raw data, but itstill relies on experiencefor its formation. Thus, mathematics is not a mere convention, but a mental organization of data in accordance with its observable features.
That is to say: without humans (or the like), events like “one plus one equals two” would not actually arise in the world; yet humans cannot say “one plus one equals three” without thereby contradicting their experience. Thus, mathematics is a product of the experience and understanding of human beings, and not some wild fantasy of theirs.
I submit that the above presentation of numbers theory is an accurate account of how numbers actually arise in the minds of individuals, and how they actually arose in the course of human history. It is not intended as a mathematician’s description of events, or a psychologist’s, but as an epistemological account. The latter is more fundamental, and fits numerical concepts in with non-numerical ones in the wider theory of knowledge.
Some comments on “modern maths”.I say this of course with “mathematical logic” in mind. In modern times, starting in the 19thCentury, attempts were made to logically streamline arithmetic, better explain it and fit it in the larger context of our knowledge. This is naturally a worthy undertaking, but some erroneous presuppositions were made in the course of it and some unjustifiable inferences were drawn from some of the results obtained.
We may here mention, as a major example, the work ofGottlob Frege. His goal was to make more explicit all the logical steps involved in the development of arithmetic. But in pursuing this commendable goal, he thought he was engaged in purely formal and deductive acts, and did not realize the extent to which he was actually depending on experience and conceptual insight.
This is evident for example in the way he developed cardinal numbers, defining them by reference to disjunctions of members of a set (if I am not mistaken). A set with one disjunction would have two members, one with two disjunctions would have three members, and so forth. Now, this is ingenious, and seems to reduce numerical development to a series of purely logical statements, relying only on abstract concepts like identity and difference, sets and members, and disjunction (i.e. negation of conjunction).
But we can ask many questions. First, does this way of presenting things correspond to the way humans actually conceptualize numbers, as individuals and collectively in history? The answer is, I suggest, no. Frege’s system may well be interesting to logicians and mathematicians as an abstract ex post facto ordering of mathematical knowledge acquired till then, and as a way to develop new mathematical knowledge, but it is essentially an artificial and recent construction.
It is more an abstract game than a description of how human knowledge of numbers occurs in practice. Such a construct cannot be said to radically discredit and displace the arithmetic notions that precede it. This would be committing the genetic fallacy, i.e. forgetting the debt owed to what came before in the human mind. It is only because we have already assimilated numbers that we are now able to play around with them the way Frege does.
Secondly, to fully understand this, and agree with it, consider when, where and how the concepts Frege uses in his system arise. It is a misrepresentation to think that he is functioning on some entirely abstract and mechanical plane, without appeal to experience or inductive reasoning. Does he anywhere reflect on how we have knowledge of identity and difference, sets and members, and disjunction (i.e. negation of conjunction), in specific situations and in general?
If we do reflect on these issues, a bit more deeply than Frege ever did, we quickly realize that these concepts are not so simple and primary. Identity and difference involve the cognitive acts of comparison and contrast, which rely on experience and on its subdivision. This indeed occurs before the number one (1) is first grasped, as above mentioned; but the point made here is that it is not as instantaneous and mechanical as Frege imagines.
The concepts of sets and members, and even that of negation, are effectively taken by him as primaries, whereas they require much, much study and reflection to understand and use. It is true that children routinely grasp them enough to use them, but that only goes to show the native intelligence of the human species. In truth, if we want to “formalize” their thinking in the way Frege tries, we have to first clarify the genesis of such abstractions in detail.
Why, for instance, refer to disjunction, when conjunction precedes it in knowledge? We only understand disjunction by negation of conjunction. So to explain numbers through conjunction as above proposed, is the more natural way – and this is the way real humans proceed. I do not pretend to be a mathematician, but I feel safe in saying that a mathematical system based on natural numbers developed by successive conjunction of units is just as good, if not better than Frege’s contraption.
It does not follow from such reflections that mathematics is something psychological, i.e. that we need to study psychology to get to mathematics, or anything of the sort. What it does mean is that if you study “mathematical logic” without asking and answering the deeper epistemological questions, all you will have is an abstract construct.
You should not, thereafter, boast to have done away with or replaced epistemology; all you have done is ignored it. You have flexed your muscles in manipulation of symbols, but you have not demonstrated insight into how you actually functioned while doing so. Your “proofs” are then just superficial processes, which do not take into account every assumption hidden within them.
This is not intended to mean that so-called mathematical logic is not objective, i.e. is necessarily divorced from reality in some significant way. What it is intended to mean is that such studies occur in a sandbox, without proper awareness of the wider world.
See also my earlier comments on this topic, inPhenomenology(II.4 and VIII) and inRuminations(9.10).
And of course, even though, the things we may have in mind behind the figures, e.g. apples and oranges, may be different in various respects.
To eat it, or give it or trade it.
When I say mentally, I mean by projecting divisions and groupings; i.e. imagination or even hallucination is used.
Germany, 1848-1925.Though Frege did not himself draw any larger philosophical conclusions from his results, his contemporary Bertrand Russell did so.