Reductionism of Numerical Complexity: A Wittgensteinian Excursion

Wittgenstein’s criticism of Russell’s logicist foundation of mathematics contained in (Remarks on the Foundation of Mathematics) consists in saying that it is not the formalized version of mathematical deduction which vouches for the validity of the intuitive version but conversely.

If someone tries to shew that mathematics is not logic, what is he trying to shew? He is surely trying to say something like: If tables, chairs, cupboards, etc. are swathed in enough paper, certainly they will look spherical in the end.

He is not trying to shew that it is impossible that, for every mathematical proof, a Russellian proof can be constructed which (somehow) ‘corresponds’ to it, but rather that the acceptance of such a correspondence does not lean on logic.

Taking up Wittgenstein’s criticism, Hao Wang (Computation, Logic, Philosophy) discusses the view that mathematics “is” axiomatic set theory as one of several possible answers to the question “What is mathematics?”. Wang points out that this view is epistemologically worthless, at least as far as the task of understanding the feature of cognition guiding is concerned:

Mathematics is axiomatic set theory. In a definite sense, all mathematics can be derived from axiomatic set theory. [ . . . ] There are several objections to this identification. [ . . . ] This view leaves unexplained why, of all the possible consequences of set theory, we select only those which happen to be our mathematics today, and why certain mathematical concepts are more interesting than others. It does not help to give us an intuitive grasp of mathematics such as that possessed by a powerful mathematician. By burying, e.g., the individuality of natural numbers, it seeks to explain the more basic and the clearer by the more obscure. It is a little analogous to asserting that all physical objects, such as tables, chairs, etc., are spherical if we swathe them with enough stuff.

Reductionism is an age-old project; a close forerunner of its incarnation in set theory was the arithmetization program of the 19th century. It is interesting that one of its prominent representatives, Richard Dedekind (Essays on the Theory of Numbers), exhibited a quite distanced attitude towards a consequent carrying out of the program:

It appears as something self-evident and not new that every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers [ . . . ] But I see nothing meritorious [ . . . ] in actually performing this wearisome circumlocution and insisting on the use and recognition of no other than rational numbers.

Perec wrote a detective novel without using the letter ‘e’ (La disparition, English A void), thus proving not only that such an enormous enterprise is indeed possible but also that formal constraints sometimes have great aesthetic appeal. The translation of mathematical propositions into a poorer linguistic framework can easily be compared with such painful lipogrammatical exercises. In principle all logical connectives can be simulated in a framework exclusively using Sheffer’s stroke, and all cuts (in Gentzen’s sense) can be eliminated; one can do without common language at all in mathematics and formalize everything and so on: in principle, one could leave out a whole lot of things. However, in doing so one would depart from the true way of thinking employed by the mathematician (who really uses “and” and “not” and cuts and who does not reduce many things to formal systems). Obviously, it is the proof theorist as a working mathematician who is interested in things like the reduction to Sheffer’s stroke since they allow for more concise proofs by induction in the analysis of a logical calculus. Hence this proof theorist has much the same motives as a mathematician working on other problems who avoids a completely formalized treatment of these problems since he is not interested in the proof-theoretical aspect.

There might be quite similar reasons for the interest of some set theorists in expressing usual mathematical constructions exclusively with the expressive means of ZF (i.e., in terms of ∈). But beyond this, is there any philosophical interpretation of such a reduction? In the last analysis, mathematicians always transform (and that means: change) their objects of study in order to make them accessible to certain mathematical treatments. If one considers a mathematical concept as a tool, one does not only use it in a way different from the one in which it would be used if it were considered as an object; moreover, in semiotical representation of it, it is given a form which is different in both cases. In this sense, the proof theorist has to “change” the mathematical proof (which is his or her object of study to be treated with mathematical tools). When stating that something is used as object or as tool, we have always to ask: in which situation, or: by whom.

A second observation is that the translation of propositional formulæ in terms of Sheffer’s stroke in general yields quite complicated new formulæ. What is “simple” here is the particularly small number of symbols needed; but neither the semantics becomes clearer (p|q means “not both p and q”; cognitively, this looks more complex than “p and q” and so on), nor are the formulæ you get “short”. What is looked for in this case, hence, is a reduction of numerical complexity, while the primitive basis attained by the reduction cognitively looks less “natural” than the original situation (or, as Peirce expressed it, “the consciousness in the determined cognition is more lively than in the cognition which determines it”); similarly in the case of cut elimination. In contrast to this, many philosophers are convinced that the primitive basis of operating with sets constitutes really a “natural” basis of mathematical thinking, i.e., such operations are seen as the “standard bricks” of which this thinking is actually made – while no one will reasonably claim that expressions of the type p|q play a similar role for propositional logic. And yet: reduction to set theory does not really have the task of “explanation”. It is true, one thus reduces propositions about “complex” objects to propositions about “simple” objects; the propositions themselves, however, thus become in general more complex. Couched in Fregean terms, one can perhaps more easily grasp their denotation (since the denotation of a proposition is its truth value) but not their meaning. A more involved conceptual framework, however, might lead to simpler propositions (and in most cases has actually just been introduced in order to do so). A parallel argument concerns deductions: in its totality, a deduction becomes more complex (and less intelligible) by a decomposition into elementary steps.

Now, it will be subject to discussion whether in the case of some set operations it is admissible at all to claim that they are basic for thinking (which is certainly true in the case of the connectives of propositional logic). It is perfectly possible that the common sense which organizes the acceptance of certain operations as a natural basis relies on something different, not having the character of some eternal laws of thought: it relies on training.

Is it possible to observe that a surface is coloured red and blue; and not to observe that it is red? Imagine a kind of colour adjective were used for things that are half red and half blue: they are said to be ‘bu’. Now might not someone to be trained to observe whether something is bu; and not to observe whether it is also red? Such a man would then only know how to report: “bu” or “not bu”. And from the first report we could draw the conclusion that the thing was partly red.

Frege-Russell and Mathematical Identity

Frege considered it a principal task of his logical reform of arithmetic to provide absolutely determinate identity conditions for the objects of that science, i.e. for numbers. Referring to the contemporary situation in this discipline he writes:

How I propose to improve upon it can be no more than indicated in the present work. With numbers … it is a matter of fixing the sense of an identity.

Frege makes the following critically important assumption : identity is a general logical concept, which is not specific to mathematics. Frege says:

It is not only among numbers that the relationship of identity is found. From which it seems to follow that we ought not to define it specially for the case of numbers. We should expect the concept of identity to have been fixed first, and that then from it together with the concept of number it must be possible to deduce when numbers are identical with one another, without there being need for this purpose of a special definition of numerical identity as well.

In a different place Frege says clearly that this concept of identity is absolutely stable across all possible domains and contexts:

Identity is a relation given to us in such a specific form that it is inconceivable that various forms of it should occur.

Frege’s definition of natural number, as modified in Russell (Bertrand Russell – Principles of Mathematics) later became standard. Intuitively the number 3 is what all collections consisting of three members (trios) share in common. Now instead of looking for a common form, essence or type of trios let us simply consider all such things together. According to Frege and Russell the collection (class, set) of all trios just is the number 3. Similarly for other numbers. Isn’t this construction circular? Frege and Russell provide the following argument which they claim allows us to avoid circularity here: given two different collections we may learn whether or not they have the same number of members without knowing this number and even without the notion of number itself. It is sufficient to find a one-one correspondence between members of two given collections. If there is such a correspondence, the two collections comprise the same number of members, or to avoid any reference to numbers we can say that the two collections are equivalent. This equivalence is Humean. Let us define natural numbers as equivalence classes under this relation. This definition reduces the question of identity of numbers to that of identity of classes. This latter question is settled through the axiomatization of set theory in a logical calculus with identity. Thus Frege’s project is realized: it has been seen how the logical concept of identity applies to numbers. In an axiomatic setting “identities” in Quine’s sense (that is, identity conditions) of mathematical objects are provided by an axiom schema of the form

∀x ∀y (x=y ↔ ___ )

called the Identity Schema (IS). This does not resolve the identity problem though because any given system of axioms, generally speaking, has multiple models. The case of isomorphic models is similar to that of equal numbers or coincident points (naively construed): there are good reasons to think of isomorphic models as one and there is also good reason to think of them as many. So the paradox of mathematical “doubles” reappears. It is a highly non-trivial fact that different models of Peano arithmetic, ZF, and other important axiomatic systems are not necessarily isomorphic. Thus logical analysis à la Frege-Russell certainly clarifies the mathematical concepts involved but it does not settle the identity issue as Frege believed it did. In the recent philosophy of mathematics literature the problem of the identity of mathematical objects is usually considered in the logical setting just mentioned: either as the problem of the non-uniqueness of the models of a given axiomatic system or as the problem of how to fill in the Identity Schema. At the first glance the Frege-Russell proposal concerning the identity issue in mathematics seems judicious and innocent (and it certainly does not depend upon the rest of their logicist project): to stick to a certain logical discipline in speaking about identity (everywhere and in particular in mathematics).

Geometric Structure, Causation, and Instrumental Rip-Offs, or, How Does a Physicist Read Off the Physical Ontology From the Mathematical Apparatus?

The benefits of the various structuralist approaches in the philosophy of mathematics is that it allows both the mathematical realist and anti-realist to use mathematical structures without obligating a Platonism about mathematical objects, such as numbers – one can simply accept that, say, numbers exist as places in a larger structure, like the natural number system, rather than as some sort of independently existing, transcendent entities. Accordingly, a variation on a well-known mathematical structure, such as exchanging the natural numbers “3” and “7”, does not create a new structure, but merely gives the same structure “relabeled” (with “7” now playing the role of “3”, and visa-verse). This structuralist tactic is familiar to spacetime theorists, for not only has it been adopted by substantivalists to undermine an ontological commitment to the independent existence of the manifold points of M, but it is tacitly contained in all relational theories, since they would count the initial embeddings of all material objects and their relations in a spacetime as isomorphic.

A critical question remains, however: Since spacetime structure is geometric structure, how does the Structural Realism (SR) approach to spacetime differ in general from mathematical structuralism? Is the theory just mathematical structuralism as it pertains to geometry (or, more accurately, differential geometry), rather than arithmetic or the natural number series? While it may sound counter-intuitive, the SR theorist should answer this question in the affirmative – the reason being, quite simply, that the puzzle of how mathematical spacetime structures apply to reality, or are exemplified in the real world, is identical to the problem of how all mathematical structures are exemplified in the real world. Philosophical theories of mathematics, especially nominalist theories, commonly take as their starting point the fact that certain mathematical structures are exemplified in our common experience, while other are excluded. To take a simple example, a large collection of coins can exemplify the standard algebraic structure that includes commutative multiplication (e.g., 2 x 3 = 3 x 2), but not the more limited structure associated with, say, Hamilton’s quaternion algebra (where multiplication is non-commutative; 2 x 3 ≠ 3 x 2). In short, not all mathematical structures find real-world exemplars (although, for the minimal nominalists, these structures can be given a modal construction). The same holds for spacetime theories: empirical evidence currently favors the mathematical structures utilized in General Theory of Relativity, such that the physical world exemplifies, say, g, but a host of other geometric structures, such as the flat Newtonian metric, h, are not exemplified.

The critic will likely respond that there is substantial difference between the mathematical structures that appear in physical theories and the mathematics relevant to everyday experience. For the former, and not the latter, the mathematical structures will vary along with the postulated physical forces and laws; and this explains why there are a number of competing spacetime theories, and thus different mathematical structures, compatible with the same evidence: in Poincaré fashion, Newtonian rivals to GTR can still employ h as long as special distorting forces are introduced. Yet, underdetermination can plague even simple arithmetical experience, a fact well known in the philosophy of mathematics and in measurement theory. For example, in Charles Chihara, an assessment of the empiricist interpretation of mathematics prompts the following conclusion: “the fact that adding 5 gallons of alcohol to 2 gallons of water does not yield 7 gallons of liquid does not refute any law of logic or arithmetic [“5+2=7”] but only a mistaken physical assumption about the conservation of liquids when mixed”. While obviously true, Chihara could have also mentioned that, in order to capture our common-sense intuitions about mathematics, the application of the mathematical structure in such cases requires coordination with a physical measuring convention that preserves the identity of each individual entity, or unit, both before and after the mixing. In the mixing experiment, perhaps atoms should serve as the objects coordinated to the natural number series, since the stability of individual atoms would prevent the sort of blurring together of the individuals (“gallon of liquid”) that led to the arithmetically deviant results. By choosing a different coordination, the mixing experiment can thus be judged to uphold, or exemplify, the statement “5+2=7”. What all of this helps to show is that mathematics, for both complex geometrical spacetime structures and simple non-geometrical structures, cannot be empirically applied without stipulating physical hypotheses and/or conventions about the objects that model the mathematics. Consequently, as regards real world applications, there is no difference in kind between the mathematical structures that are exemplified in spacetime physics and in everyday observation; rather, they only differ in their degree of abstractness and the sophistication of the physical hypotheses or conventions required for their application. Both in the simple mathematical case and in the spacetime case, moreover, the decision to adopt a particular convention or hypothesis is normally based on a judgment of its overall viability and consistency with our total scientific view (a.k.a., the scientific method): we do not countenance a world where macroscopic objects can, against the known laws of physics, lose their identity by blending into one another (as in the addition example), nor do we sanction otherwise undetectable universal forces simply for the sake of saving a cherished metric.

Another significant shared feature of spacetime and mathematical structure is the apparent absence of causal powers or effects, even though the relevant structures seem to play some sort of “explanatory role” in the physical phenomena. To be more precise, consider the example of an “arithmetically-challenged” consumer who lacks an adequate grasp of addition: if he were to ask for an explanation of the event of adding five coins to another seven, and why it resulted in twelve, one could simply respond by stating, “5+7=12”, which is an “explanation” of sorts, although not in the scientific sense. On the whole, philosophers since Plato have found it difficult to offer a satisfactory account of the relationship between general mathematical structures (arithmetic/”5+7=12”) and the physical manifestations of those structures (the outcome of the coin adding). As succinctly put by Michael Liston:

Why should appeals to mathematical objects [numbers, etc.] whose very nature is non-physical make any contribution to sound inferences whose conclusions apply to physical objects?

One response to the question can be comfortably dismissed, nevertheless: mathematical structures did not cause the outcome of the coin adding, for this would seem to imply that numbers (or “5+7=12”) somehow had a mysterious, platonic influence over the course of material affairs.

In the context of the spacetime ontology debate, there has been a corresponding reluctance on the part of both sophisticated substantivalists and (R2, the rejection of substantivalist) relationists to explain how space and time differentiate the inertial and non-inertial motions of bodies; and, in particular, what role spacetime plays in the origins of non-inertial force effects. Returning once more to our universe with a single rotating body, and assuming that no other forces or causes, it would be somewhat peculiar to claim that the causal agent responsible for the observed force effects of the motion is either substantival spacetime or the relative motions of bodies (or, more accurately, the motion of bodies relative to a privileged reference frame, or possible trajectories, etc.). Yet, since it is the motion of the body relative to either substantival space, other bodies/fields, privileged frames, possible trajectories, etc., that explains (or identifies, defines) the presence of the non-inertial force effects of the acceleration of the lone rotating body, both theories are therefore in serious need of an explanation of the relationship between space and these force effects. The strict (R1) relationists face a different, if not less daunting, task; for they must reinterpret the standard formulations of, say, Newtonian theory in such a way that the rotation of our lone body in empty space, or the rotation of the entire universe, is not possible. To accomplish this goal, the (R1) relationist must draw upon different mathematical resources and adopt various physical assumptions that may, or may not, ultimately conflict with empirical evidence: for example, they must stipulate that the angular momentum of the entire universe is 0.

All participants in the spacetime ontology debate are confronted with the nagging puzzle of understanding the relationship between, on the one hand, the empirical behavior of bodies, especially the non-inertial forces, and, on the other hand, the apparently non-empirical, mathematical properties of the spacetime structure that are somehow inextricably involved in any adequate explanation of those non-inertial forces – namely, for the substantivalists and (R2) relationists, the affine structure,  that lays down the geodesic paths of inertially moving bodies. The task of explaining this connection between the empirical and abstract mathematical or quantitative aspects of spacetime theories is thus identical to elucidating the mathematical problem of how numbers relate to experience (e.g., how “5+7=12” figures in our experience of adding coins). Likewise, there exists a parallel in the fact that most substantivalists and (R2) relationists seem to shy away from positing a direct causal connection between material bodies and space (or privileged frames, possible trajectories, etc.) in order to account for non-inertial force effects, just as a mathematical realist would recoil from ascribing causal powers to numbers so as to explain our common experience of adding and subtracting.

An insight into the non-causal, mathematical role of spacetime structures can also be of use to the (R2) relationist in defending against the charge of instrumentalism, as, for instance, in deflecting Earman’s criticisms of Sklar’s “absolute acceleration” concept. Conceived as a monadic property of bodies, Sklar’s absolute acceleration does not accept the common understanding of acceleration as a species of relative motion, whether that motion is relative to substantival space, other bodies, or privileged reference frames. Earman’s objection to this strategy centers upon the utilization of spacetime structures in describing the primitive acceleration property: “it remains magic that the representative [of Sklar’s absolute acceleration] is neo-Newtonian acceleration

d2xi/dt2 + Γijk (dxj/dt)(dxk/dt) —– (1)

[i.e., the covariant derivative, or ∇ in coordinate form]”. Ultimately, Earman’s critique of Sklar’s (R2) relationism would seem to cut against all sophisticated (R2) hypotheses, for he seems to regard the exercise of these richer spacetime structures, like ∇, as tacitly endorsing the absolute/substantivalist side of the dispute:

..the Newtonian apparatus can be used to make the predictions and afterwards discarded as a convenient fiction, but this ploy is hardly distinguishable from instrumentalism, which, taken to its logical conclusion, trivializes the absolute-relationist debate.

The weakness of Earman’s argument should be readily apparent—since, to put it bluntly, does the equivalent use of mathematical statements, such as “5+7=12”, likewise obligate the mathematician to accept a realist conception of numbers (such that they exist independently of all exemplifying systems)? Yet, if the straightforward employment of mathematics does not entail either a realist or nominalist theory of mathematics (as most mathematicians would likely agree), then why must the equivalent use of the geometric structures of spacetime physics, e.g., ∇ require a substantivalist conception of ∇ as opposed to an (R2) relationist conception of ∇? Put differently, does a substantivalist commitment to whose overall function is to determine the straight-line trajectories of Neo-Newtonian spacetime, also necessitate a substantivalist commitment to its components, such as the vector d/dt along with its limiting process and mapping into ℜ? In short, how does a physicist read off the physical ontology from the mathematical apparatus? A non-instrumental interpretation of some component of the theory’s quantitative structure is often justified if that component can be given a plausible causal role (as in subatomic physics)—but, as noted above, ∇ does not appear to cause anything in spacetime theories. All told, Earman’s argument may prove too much, for if we accept his reasoning at face value, then the introduction of any mathematical or quantitative device that is useful in describing or measuring physical events would saddle the ontology with a bizarre type of entity (e.g., gross national product, average household family, etc.). A nice example of a geometric structure that provides a similarly useful explanatory function, but whose substantive existence we would be inclined to reject as well, is provided by Dieks’ example of a three-dimensional colour solid:

Different colours and their shades can be represented in various ways; one way is as points on a 3-dimensional colour solid. But the proposal to regard this ‘colour space’ as something substantive, needed to ground the concept of colour, would be absurd.

Neo-Kantians and Numbers. Note Quote.

At the beginning of the twentieth century, neo-Kantianism was the dominant force in German academic philosophy. Its most important schools were Marburg and Southwestern (or Baden). The Marburg school concentrated on logical, methodological and epistemological themes. Its founder and leader was Hermann Cohen (1842-1918), a professor of philosophy at Marburg between 1876 and 1912. Cohen’s most famous disciples were Paul Natorp (1854-1924) and Ernst Cassirer (1874-1945). The Southwestern school emphasised the theory of values. Its founder and leader was Wilhelm Windelband (1848-1915). Windelband’s student Heinrich Rickert (1863-1936) was the great system-builder of the Southwestern school. Among the members of the Southwestern school were Jonas Cohn (1869-1947) and Bruno Bauch (1877-1942). At the beginning of the twentieth century, the philosophy of mathematics in general and the nature of number in particular were subjects of lively discussion among the neo-Kantians. Natorp, Cassirer and Cohn, among others, constructed their own theories of number which also formed the basis of their critiques of Russell and Frege. The neo-Kantians, too, supported the idea that mathematics should be based on a logical foundation. However, their conception of the logical foundation differs greatly from that of Russell and Frege. The main difference is that although the neo-Kantians argue that mathematics should be based on a logical foundation, these two sciences must be strictly separated from one another. Consequently, they argued that if the logicist programme were carried out, there would not exist any line of demarcation between logic and mathematics. Cohn’s distinction between two possible ways to found the number concept on logic brings forward the main difference between Russell’s and the neo-Kantians’ viewpoint. According to Cohn, there exist two possible ways to found the number concept on logic. Either the number concept is reduced to a logical concept or it is shown that the number concept itself is a fundamental logical concept. Cohn says that while Russell’s theory of number is founded on logic in the first sense, his own theory of number is logical in the latter sense.

According to the neo-Kantians, deducing the number concept from the class concept is a petitio principii. In other words, Cassirer, Natorp and Cohn all argue that the class concept already presupposes the number concept. In Cassirer’s own words,

What it means to apprehend an object as “one” is here assumed to be known from the very beginning; for the numerical equality of two classes is known solely by the fact that we coordinate with each element of the first class one and only one of the second.

According to Cohn, the number concept is something that logically precedes the class concept. As Cohn sees it, Russell’s definitions often contain such expressions as “an object” and Russell himself admits that the sense in which every object is one is always involved when speaking of an object. Consequently, Cohn argues that Russell’s definition of number already presupposes the number concept. In Natorp’s view, Frege’s definition of number presupposes the use of such propositions as ‘X falls under the concept A’. As Natorp sees it, in this proposition an individual is presupposed in the sense of a singular number. Thus Frege’s definition of number already presupposes the singular number. According to Natorp, this mistake, consisting of a simple petitio principii, is shared by all attempts to derive the number concept from the concept of objects belonging to a class (or sets or aggregates). It is inevitable that these objects are thought of as individuals. It is noteworthy that Henri Poincaré presents a similar argument in his critique of the logicist programme. In his paper “Les mathématiques et la logique”  Poincaré, too, argues that the logicist definition of number already presupposes the number concept.