Mathematical Structures as Tools
In the following we will briefly review some mathematical structures with respect to their function as tools.
3.1 Natural Numbers
A Druidic myth relates how Lucanor, coming upon the other gods as they sat at the banquet table, found them drinking mead in grand style, to the effect that several were drunk, while others remained inexplicably sober; could some be slyly swilling down more than their share? The disparity led to bickering, and it seemed that a serious quarrel was brewing.
Lucanor bade the group to serenity, stating that the controversy no doubt could be settled without recourse either to blows or to bitterness. Then and there Lucanor formulated the concept of numbers and enumeration, which heretofore had not existed. The gods henceforth could tally with precision the number of horns each had consumed and, by this novel method, ensure general equity and further, explain why some were drunk and others not. “The answer, once the new method is mastered, becomes simple!” explained Lucanor. “It is that the drunken gods have taken a greater number of horns than the sober gods, and the mystery is resolved.” For this, the invention of mathematics, Lucanor was given much honor.Jack Vance, Madouc1
The natural numbers are the most basic structure in mathematics. Their original function is also clear: to count. And in this function, natural numbers are nothing like a tool. Given any collection of objects (in a non-mathematical sense: finite and consisting of physical objects) we may count its elements by attributing (consecutive) numbers to every single object, not assigning the same number to two different objects.
With the help of arithmetical and logical expressive power, we are able to develop a rather sophisticated internal structure of the natural numbers, containing an endless number of number-theoretic functions, starting with addition and multiplication, as well as relations, including the natural order relations which allow comparisons.
For such functions and relations we can express propositions which should turn out true or false. Let us look at two examples:
2+11 # 1,
(1)
(2)
ConPA
where ConPA should be a standard arithmetical consistency statement for Peano Arithmetic, given as -3x.Bew(x, r0 = 1n) with Bew the usual proof predicate from the proof of Godel’s (first) Incompleteness Theorem.
As far as (1) is concerned, nobody will have the slightest doubt that it is a true arithmetical statement. We will see below, however, in which way its truth should indeed be seen as one relative to the structure of the natural numbers. [176]
For (2) Godel’s Second Incompleteness Theorem tells us, that it is not derivable in Peano Arithmetic; still, it is definitely a true arithmetical formula.[177] Thus, it holds in the standard structure of the natural numbers. Godel’s Theorem, however, implies that there have to exist structures which satisfy the axioms of Peano Arithmetic, but this formula is false. We have also to reflect briefly on these non-standard models of PA.
Cyclic Groups
If you look at the face of your watch, do the numerals you are seeing represent natural numbers? They appear so, but, insofar counting of hours is concerned, one can clearly say that 2h after 11 o’clock is 1 o’clock. So, (1) turns out to be false, at least for the numerals of our clock-faces.
Mathematically, it is clear what happens: numbers on a clock-face are supposed to be elements of Z/12Z, the cyclic group of 12 elements, rather than to be elements of the natural numbers. In Z/12Z, of course, (1) is false. But as (1) is stated, there is nothing which forces us to read the numbers as natural numbers, and not as elements of Z/12Z. Thus, its truth is, indeed, relativized to a structure. The structure in question is almost always clear from the context; but this simple example shows that it would be misleading to consider (1) as an absolute truth.
Of course, one could try to argue that, in (1), the numbers are supposed to be natural numbers, and a reading of (1) in Z/12Z would be an abuse of notation. But such a presupposition would not turn (1) in an absolute truth; to the contrary, the very presupposition—that the numbers are supposed to be natural numbers—would just fix the relativization of (the truth of) (1) to the structure of the natural numbers. As a matter of fact, the notational “overloading” of numbers—as natural numbers or elements of a cyclic group—requires a disambiguation which just comes down to the relativization to the different structures.[178]
The clock example shows also that Z/12Z is a quite useful tool to deal with cyclic arithmetic (of period 12). Thus, there are applications where we, indeed, would like to have 2+11 = 1 to be true.
While we obtain cyclic groups quite naturally (and for any positive order) out of the natural numbers, finite fields, i.e., finite structures which are equipped with addition and multiplication, are not so easily to define. It is, in fact, a non-trivial mathematical theorem that there only finite fields of prime power order. For somebody who rejects any kind of objectivity or any notion of truth within mathematical structures, such a limitation theorem should have an objective character, at least in its negative part: there is simply no way to define a finite field of, let us say, order 6. Thus, when we say that mathematical structures are tools, this theorem gives us a formal constraint on what kind of tools are actually possible.
Insane models of PA
As said, it follows from Godel’s Incompleteness Theorems that PA is consistent together with its own formalized inconsistency statement -ConPA; aforteriori there has to be a first-order structure serving as a model for this theory. Following a terminology used by Kikuchi and Kurahashi (2016), one can call such a model insane. However, there is a way to understand what is going on in such an insane model— Takeuti reports how he learned it from Godel (Yasugi and Passell 2003, p.
3):[Godel’s] way of teaching nonstandard models was an interesting one. It went as follows. Let T be a theory with a nonstandard model. By virtue of his Incompleteness Theorem, the consistency proof of T cannot be carried out within T. Consequently, T and the proposition “T is inconsistent” is consistent. There is, therefore, a natural number N which is the Godel number of the proof leading to a contradiction from T. Such a number is obviously an infinite natural number.
In an insane model of PA it is indeed the case that the inconsistency statement of PA is true. But we do not run into a contradiction: the formula -ConPA has received a new meaning which simply does not deserve to be called an “inconsistency statement” any longer.[179]
According to our line of argument, insane models of PA will be tools in mathematics in the same way as the standard structure of the natural numbers. And yes, they are a tool, but, admittedly utterly useless tools: at least to our knowledge there is not a single mathematical problem which would profit from being treated by an insane model of PA.
So, again, the truth of (2) holds only in the (intended) standard structure of the natural numbers; the fact that there are non-standard models of PA in which (2) is false just requires that we have to indicate which structure is under consideration.
3.2 Geometry
As discussed above, the discovery of non-Euclidean Geometry led, eventually, to the conclusion that one cannot speak about truth simpliciter in the case of the Parallel Postulate. Experience shows that the different Geometries have all their raison d’etre; and they function as different tools for different purposes.
Even long before the discovery of non-Euclidean Geometry, there existed another example which shows how Geometries functions as a tool: as the name suggests, Spherical Geometry is just the adequate tool for geometric reasoning on a sphere. And it is a useful tool even in cases where no physical sphere is present: its very conception was triggered by the aim to study the stellar sphere—which, as we know now, is anything but a sphere.
3.3 Set Theory
Cantor designed set theory as a tool to give a proper fundation to the real numbers and the concept of function.[180] For this purpose, it is somehow another type of tool as the Natural Numbers and Geometry, as it is not used for applications outside of Mathematics, but rather as a tool within Mathematics. But as such, it turned out that it can be considered as a kind of universal tool, allowing for an encoding of essentially every other mathematical structure.[181]
In its function as a universal mathematical base, and with the appearance of the set-theoretical paradoxes, the notion of truth for set theory gains additional importance. But, as discussed above, we may consider different set-theoretic universes. One example, already mentioned, is Godel’s constructible hierarchy L, which restricts the power set operation to definable subsets. As we would like to have in our standard set-theoretic universe more subsets in a power set, L is generally not consider to be the “true set theory”. Still, it has turned out to be an extraordinary useful tool in Mathematics (first of all, in Godel’s proof of the consistency of CH with ZFC); a tool we would not like to deprive ourselves of, in particular not on the basis of the argument that it is not our intended “true set theory”.
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