On ‘The Unreasonable Effectiveness of Mathematics’ and the Indispensability Response

There are, on the other hand, aspects of the world concerning which we do not believe in the existence of any accurate regularities. We call these initial conditions.

EUGENE WIGNER

This essay seeks to contribute to the literature by revisiting the long-standing debate concerning the ‘unreasonable effectiveness of mathematics’, as first presented by Eugene Wigner. It draws upon the indispensability argument as a possibly reliable and well suited response, while noting that there are many dimensions to Wigner’s open-ended puzzle, and that not all of these lend themselves to a clear appraisal by the analytic-minded philosophy of the indispensability argument. The unconventional thesis put forward is that the indispensability argument falls short of expectations given its difficulty in accounting for the initial conditions. The paper begins by outlining Wigner’s ‘unreasonable effectiveness’ problem and the indispensability argument(s). The responses of the latter are then examined, before a model of the issues is introduced, and its attendant implications are conveyed.

1. Wigner’s challenge

Wigner lays out a series of problems associated with the successful applicability of mathematics in science and raises the grave concern that this puts into doubt the uniqueness of our physical theories. Wigner’s starting point consists in the miracles arising within mathematics and physics, which can be restated as the miracle of man’s capacity to divine the ‘laws of nature’, and the miracle that they should exist at all. Then, there are the miracles of mathematics and physics, expressed in the intertemporal, alignment, and autonomy problems. The intertemporal problem is that physicists do not just shop for mathematical theories but develop them independently and then recognise they were ‘conceived before by the mathematician’ (Wigner 1960: 07). The alignment problem is that for science, mathematics is not just ‘the only language we can speak’, i.e. simple, but it is indeed ‘the correct language’ (ibid.: 08), i.e. true. Finally, the autonomy problem (or the ‘empirical law of epistemology’) is that once intuited, a theory will function before all work is completed and initial conditions are understood, so that they are underdetermined by the mathematical apparatus.

2. The metaphysical (intertemporal/autonomy) problem: Quine’s response

It will be helpful to begin with the indispensability argument’s response to the metaphysical (intertemporal and autonomy) problems first before examining the semantic (alignment) problem second. Recall the metaphysical version (Quine 1980: 291) states:

P1. We ought to have ontological commitment to all and only those entities that are indispensable to our current best scientific theories;

P2. Mathematical entities are indispensable to our current best scientific theories;

Conc. We ought to have ontological commitment to (some) mathematical entities.

Given this, Quine (1980: 291) may have a response to Wigner: the ‘miracles’ of intertemporal applicability and autonomous discovery are not really miracles but realisations of a much simpler kind. These occurrences follow from the fact, guaranteed by their indispensability, that the entities underpinning them exist (or so we should ontologically commit to believe). If Wigner were a realist, perhaps he would not be so surprised. The issue at hand, however, is more complex than this. Firstly, this manoeuvre is exposed to nominalist accounts which feature empirically equivalent theories that do not quantify over the entities in question. Secondly, the nature of Wigner’s argument extends in any case beyond whether mathematical entities truly exist: this is an issue of whether their indispensability allows them to guide and develop science independently of the latter’s discoverers. These are two key vulnerabilities.

2(a). First way to proceed

Clearly the argument is not fit to address Wigner’s concerns as it stands. For its proponents, there are two ways to proceed. Firstly, they could double down on the logical form of the argument and defend its premises, resolving the first vulnerability. This is achieved by showing that the empirically equivalent theories that do not quantify over the entities in question will be dispensable. Recall for Quine, an entity ξ appearing in T can be said to be indispensable only when its elimination from T produces (i) a new theory  that is equivalent but (ii) otherwise less preferable to T. Preference in this sense implies Quine’s theory of regimentation, but this is susceptible to attack given its circular implications. We should adopt ‘the simplest conceptual scheme’ (Quine 1961: 16) and choose ‘with a view to simplicity in one’s overall system of the world’ (ibid. 1981: 10). However, scientific theories are often rendered simple by their being mathematical. So the mathematical grounding of a theory weighs in our deciding, on the basis of simplicity, between a mathematical theory and a nominalised one, which delivers—in the absence of any justified separation between mathematics and simplicity—a circular if not predeterminate conception of indispensability.

black mathematics board with formulas
Picture 1 Pi, a transcendental number, is widely recognised as one of the most pervasive and remarkably accurate mathematical entities used to describe natural phenomena

2(b). Second way to proceed

The second option available to the argument’s proponents is to draw upon a causal platonist account of mathematics, link its logical form to the indispensability argument, and thereby resolve the second vulnerability. This is achieved by first observing, for instance, that Newton believed in the existence of a gravitational law to explain (among other things) certain facts about the motion of the planets, despite little evidence in favour of it, not to mention the fact it was ‘repugnant to his time and to himself’ (Wigner 1960: 08). It was no miracle that the law of gravitation guided and developed Newton’s ideas independently of himself (or at least what he found to be not repugnant), given it exists and was causally responsible for such a development. On this view we could suppose the same for mathematical entities. Hence, hypothesising a common logical form, namely, that ‘if apparent reference to some entity (or class of entities) ξ is indispensable to our best scientific theories, then we ought to believe in the existence of ξ’ (Colyvan 2001a: 07). Notwithstanding the improbability of such a link obtaining (see Maddy (1992: 60)), such an attempt, even if it were to succeed, would conceivably struggle to explain why mathematical entities in particular are causally, as opposed to referentially, responsible. Recall that before we can explore Quine’s possible response to Wigner’s metaphysical questions, the two key vulnerabilities must be resolved. In this light neither seem like they could be. However, there is one last alternative: the explanatory defence.

2(c). The metaphysical problem: explanatory indispensability response

The metaphysical problems raised by Wigner—intertemporal applicability and autonomous discovery—could be, after all, reducible to the explanatorily indispensable role mathematical entities play in our best scientific theories, demanding a reformulation of the following character:

P1. We ought to have ontological commitment to all and only those entities that are indispensably explanatory in our current best scientific theories;

P2. Mathematical entities are indispensably explanatory in our current best scientific theories;

Conc. We ought to have ontological commitment to (some) mathematical entities.

This account (Baker 2009: 613) appears to resolve the two vulnerabilities. Firstly, the nominalist accounts can be confronted without appeal to circularity by virtue of two tests: a ‘replacement’ and ‘comparison’ test. The former entails eliminating the mathematical claim from the explanation and then judging it to be (constitutively speaking) mathematically non-eliminable if the explanatory value subsequently disappears. The second test compares all explanations and determines that mathematics is explanatorily indispensable if the chosen account has (i) the greatest explanatory power and (ii) is mathematical. This account can be used to justify the role of mathematics in guiding and developing science independently of the latter’s discoverers by showing that the relevant entities are explanatorily indispensable (Ginammi 2016: 65). The problem, however, is that for this metaphysical response to proceed, a further move is required. This argument is only powerful enough to defend mathematical claims, not entities. Furthermore, the two tests eliminate circularity at the expense of a full description of the issues: mathematics is explanatorily indispensable because it provides for the greatest explanatory power, but no reason is given for why this is the case.

3. The semantic (alignment) problem: Putnam’s response

Given this state of affairs, one may conclude that it has been very difficult to respond to Wigner’s metaphysical questions because they demand justifying mathematical entities as opposed to claims. However, the thesis to be shown suggests that this issue runs deeper: it is not a matter of entities or claims, much less realism or anti-realism. Rather, it a symptom of the irregularities of the world, the initial conditions. To demonstrate the point, it will be helpful to turn to the semantic (alignment) problem, that is, that mathematics, the language of science, aligns mysteriously with the truth. Putnam’s (1979: 338) version of the indispensability argument offers a possible response, stipulating that:

P1. We ought to believe in the truth of all and only those claims that are indispensable to our current best scientific theories;

P2. Mathematical claims are indispensable to our current best scientific theories;

Conc. We ought to believe in the truth of (some) mathematical claims.

Crucially, Putnam’s definition of indispensability differs from Quine’s, who subjects scientists’ use of mathematical claims to philosophical regimentation. Putnam takes them at face value, believing they should be respected by philosophers. Perhaps mathematical descriptions align with the truth because the claims from which they derive are true. This, however, belies Quine and Putnam’s profound difficulties with the initial conditions: given we develop science in isolation from the ‘baffling complexity’ of the world (Wigner 1960: 02), why are mathematical truths in particular indispensable to science? Why something rather than everything? The ‘narrow selection’ (ibid. 1960: 01) we make appears to have far-reaching consequences for the formulation of our best scientific theories, opening a circular reflexivity in the argument.

3(a). The semantic problem: explanatory indispensability response

One could argue mathematical claims in particular mediate the truth because they are (i) true and (ii) explanatorily indispensable. However, this does not explain why mathematics should succeed in conveying science truthfully as well as elegantly. In disciplines such as economics, mathematics typically expresses simplicity at the expense of truth (e.g. Keynesian income-expenditure model vs. national accounts) and truth at the expense of simplicity (e.g. stock-flow consistent models vs. dynamic stochastic general equilibrium models). It is sometimes possible, however, for mathematics to express simplicity at the expense of truth and in so doing provide a more successful account (e.g. fliud mechanics models). Such weaseling (Melia 2000: 456) undermines the indispensability argument’s semantic response by putting into doubt the referential indispensability of mathematical entities.

4. Beyond the indispensability argument: the problem of the initial conditions

One may be tempted at this juncture to conclude that these many difficulties facing the indispensability argument demonstrate its counter-intuitive unsuitability to Wigner’s puzzle. But this is to conflate its interpretative shortcomings with its methodological isolation of the initial conditions. By examining a model formalising the responses rehearsed above, it will be easier to draw out the connections between Wigner’s puzzle and the indispensability argument that could not otherwise obtain. In particular, it will be helpful to plunge into the realm of the metaphysical, the many strands of the argument. The model presented in Figure 1 can be understood at two levels of analysis, the first being reified, and the other abstracted. This should be understood separately from the spectrum denoted by the axes, which convey increasing reification to the right and above, and increasing abstraction to the left and below. Quadrant I models the premises of the indispensability argument, i.e. the applicability of mathematics; quadrant IV, its conclusion, the derivation of mathematical entities; quadrant III, Wigner’s puzzle, the cosmos coming to know itself via mind; and quadrant II, Wigner’s conclusion, the emergence of ready-to-wear mathematics.

Screen Shot 2018-07-24 at 5.55.37 pm.png
Figure 1 Initial conditions model, featuring {math, science}, {science, metaphysics}, {metaphysics, art}, and {art, mathematics} matrices.

4(a). Quadrant I: Mathematics and science (the indispensability argument)

In Quadrant I, looking at the curve of low scientific literacy, SL, it can be seen that as applicability of mathematics increases, the movement (tracked by the orange parabola) to our best scientific theories, Tbest, accelerates disproportionately until beyond a certain level (red line) of mathematical indispensability, , we begin to ‘“[get] something out” of the equations that we did not put in’ (Wigner 1960: 14). This conveys the autonomy problem, a process that continues until the laws of nature ‘asymptotically approach such a fusion’ (ibid. 1960: 12) as the theories of Newton and Einstein did for instance, at the level governed by the contemporary state of mathematics (dashed red line), before jumping to a higher curve. The instantiations shown are therefore only three of infinitely many possible asymptotic pairings.

4(b). Quadrant IV: Science and metaphysics (derivation of metaphysical entities)

Quadrant IV charts the derivation of the mathematical entities (claims) to which we are ontologically committed (believe in). These value judgements can be combined, for they bear the same implication in the abstract, namely, realism. Objects derived as a function of indispensability number very little from mediocre scientific theories. As best scientific theories are broached, all objects collapse into the point shown, specifying –total metaphysical items. The curve’s structure illustrates three key points: theories may quantify over more objects than that number of entities/claims deemed indispensable; quantities are governed by an inconstant load, as best science changes; and entities emergent in our best science are used to conjure other (mediocre) worlds that are different from, if not contradictory to, our world, implying that they should exceed in number those applications we deem successful and true. Taken together, these forces highlight the underdetermination of our physical theories by mathematics (Gelfert 2014: 998), questioning their uniqueness.

4(c). Quadrant III: Metaphysics and art/abstraction (meta- and paraconsistent-logic)

In the metaphysics-abstraction space these entities form a bound, Obound, above which abstractions extrapolate (note Obound is shifted down by amount b to level ). Beginning at A0, abstractions depend on innate ideas, equal to b and extrapolate up to point k as these axioms exhaust our intuitions at the level of experience, intimating Wigner’s bifurcation between elementary mathematics and complex numbers. As objects proliferate, the level of abstraction grows in super-linear scale from fixed reserve . The interplay between a priori and a posteriori abstraction plays out to the left of k, as epistemic constraints of the indispensability argument manifest, and the contingency of mathematics is implied (note variable slopes). There are three crucial points to heed. Firstly, point k specifies the initial conditions of the model; in the intermediate stage, the growth of abstraction slows as scientific complexity is integrated piece-meal; and as the level of objects approach Obound, radically emergent abstraction prevails (isomorphic to paraconsistent explosion), approximating Wigner’s ultimate concern that it is ‘quite possible that an abstract argument can be found which shows that there is a conflict between [a theory of consciousness or of biology] and the accepted principles of physics’, exposing a reflexivity between applications of mathematics and mathematics in itself, a relationship modelled in similar mapping accounts (see Bueno & Colyvan 2011: 348) stressing comparisons between mathematical structure and the target system.

4(d). Quadrant II: Art and mathematics (philosophy of mind)

Two potentialities in quadrant II interact on input from this space. The first, drawn by the curve MQ, delineates the Quinean-naturalist (‘no miracles’) intuition that as abstraction contracts (or simplicity grows) our understanding of mathematics rises. However, as unificatory simplicity peaks, mathematical knowledge cannot asymptotically develop, approaching instead a threshold, implied by Gödel’s incompleteness theorems (1931). The difference p between this limit and the upper-bound accounts for paraconsistent logic (hence the curve peaks higher than Quine’s). The second curve, MW, conveys Wigner’s argument that little mathematics proceeds from insufficient abstraction, but as the mathematician ‘exploits the domain of permissible reasoning’ (Wigner 1960: 03), incremental progress requires increasing complexity. The point at which these potentialities meet, , is at the optimum () below that level which is actually attained (), so that recreational mathematics is given by  – . But suppose outcome  prevails. On MQ, mathematicians would yield, for a relatively higher level of abstraction, few theories indispensable to best science. On MW, the level of complexity surpasses that standard required of best science. Conversely if the mathematics arose to the right of the  equilibrium, then, following Quine’s curve, an all-encompassing simplicity would provide a unified account. However, this would compel further research and abstraction, given by MW. Hence,  must specify the equilibrium.

4(e). Key insights of the model

This endogenous determination illuminates upon a number of key results. The indispensability argument appears to suggest, as mentioned above concerning regimentation, that equilibrium mathematics, , determines the point of best science . But again, there is no reason to believe these points should align. Indeed, resting at point  would be preferable as that yields complimentary science equal to j. Resting at a higher level of scientific literacy,, by contrast, would miss best science by gap h (as this demands an unattained level of simplicity). Hence, all possible science is shaded in grey, bound by the applicability-possibility frontier. The downward slope represents the notion that as the simplicity of scientific theories improves, the per unit demand for mathematical structures falls. Given these implications, it is clear the argument does not just take correspondence between  and  as brute fact (Colyvan 2001b: 271). The mathematical equilibrium is not accounted for either, because the initial conditions play no role in the argument. Therefore, a theory of the initial conditions would justify the mathematical optimum, thereby mediating abstract entities/claims with concrete theory. It is for this reason the applicability issue runs deeper: it just as much a matter of entities and claims as it is of accounting for the initial conditions, the narrow selection that continuously shapes mathematics and its applications.

5. Conclusion

It can be seen, in the final analysis, that the indispensability argument provides a reliable although vulnerable, response to Wigner’s puzzle, broadly encompassing themes of intertemporal applications, semantic alignment, and underdetermined autonomy. While Quine’s original argument presents further difficulties, subsequent and related versions of the indispensability argument, such as those of Putnam and the explanatory indispensability group, have come some way to addressing these concerns, although in some cases at the expense of a full description of the issues. In critically analysing these responses, a key issue not conventionally discussed in the literature has been proffered as a crucial link in the viability of the analysis: the initial conditions. Through a hitherto unexplored model of the issues, this feature of the background has been shown to play a crucial role in determination and concomitant applicability of mathematics.

References


Baker, A. 2009, “Mathematical Explanation In Science”, British Journal For The Philosophy of Science, 60, pp. 611–633.

Bueno, O. & Colyvan, M. 2011, ‘An Inferential Conception of the Application of Mathematics’, Noûs, vol. 45, no. 2, pp. 345-374.

Colyvan, M. 2001a. The Indispensability of Mathematics. Oxford: Oxford University Press.

Colyvan, M. 2001b. ‘The Miracle Of Applied Mathematics’, Synthese, 127, 265–277.

Gelfert, A. 2014. ‘Applicability, Indispensability, and Underdetermination: Puzzling Over Wigner’s ‘Unreasonable Effectiveness of Mathematics’’, Science & Education, vol. 23, no. 5, pp. 997-1009.

Ginammi, M. 2016. ‘The Applicability of Mathematics and the Indispensability Arguments’, Revue De La Societe De Philosophie Des Sciences, vol. 03, no. 01, pp. 59-68.

Gödel, K. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. (in Gödel, 1986), vol. 1, pp. 144–195.

Maddy, P. 1992. Realism in Mathematics. Oxford: Clarendon Press.

Melia, J. 2000. ‘Weaseling Away the Indispensability Argument’, Mind, vol. 109, no. 435 (Jul., 2000), pp. 455-479.

Putnam, Hilary. 1979a. ‘Philosophy of logic’. In Mathematics Matter and Method: Philosophical papers vol. 1. 2nd edn. Cambridge: Cambridge University Press.

Quine, W. V. 1961. From a Logical Point of View. 2nd (revised) ed. Cambridge: Harvard University Press.

Quine, W. V. 1980. ‘On What There Is’, Reprinted in From A Logical Point Of View, 2nd edition, Cambridge, MA, Harvard University Press, pp. 1–19.

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