Metaphysics, Philosophy

I. Introduction

Throughout the history of mathematical thought, the age-old philosophical debate between realism and nominalism has produced various perspectives on the ontological status of abstract entities such as functions, numbers, and sets, the most notable of which include intuitionism (or constructivism), formalism, fictionalism, and indispensabilist platonism.

According to the predominant position held by mathematical realists, which draws heavily from the indispensability argument first proposed by philosophers Willard Van Orman Quine and Hilary Putnam, the indispensability of some mathematical entities in our best scientific theories warrants belief in their existence; equivalently stated, we ought to believe in the mind-independent existence of mathematical entities whose postulation is necessary for the construction of our best scientific theories.1

In light of the instrumentalist objection to mathematical platonism, which cites the lack of ontological commitment to such idealized entities as perfectly continuous fluids and frictionless surfaces as a counterexample to the indispensability argument, an explanatoriness clause has recently been introduced in order to narrow the scope of the term “indispensable”. Hence, in his Mathematical Explanation in Science (2009), philosopher of science Alan Baker writes:

[I]t is not enough—for the purposes of establishing platonism—that mathematics be indispensable for science; it has to be indispensable in the right kind of way. Specifically, it needs to be shown that reference to mathematical objects sometimes plays an explanatory role in science.2

The purpose of this paper is to present a transcendental argument for mathematical realism in accordance with the insights provided by praxeology3; more particularly, the Kantian view of syntheticity will be integrated into an Aristotelian (rather than platonist) philosophy of mathematics in order to demonstrate that mathematical entities are realized in the external world by human action.

In so doing, I will attempt to ground mathematical realism on an incontrovertible epistemological foundation and thereby demonstrate that the Quine-Putnam indispensability argument is merely an implication of a broader, more all-encompassing justification for belief in the existence of mathematical entities; it is not our acceptance of scientific theories in which mathematical entities play an indispensable role that warrants belief in their existence but rather the necessity of affirming their existence through action that explains the success and explanatory power of our best scientific theories.

Throughout the paper, I will use the case study of the prime-numbered life cycles of periodical cicadas and the corresponding analysis provided by philosopher of science Alan Baker in order to illustrate the logical relationship between human action, mathematical entities, and physical phenomena.

II. The Metaphysics of Action

A brief overview of praxeology is prerequisite to a discussion of its relation to mathematics. The fundamental axiom of praxeology is the proposition that all people act, the concept of action here defined as purposeful behavior. Action is not a mere pursuit in which man may choose to partake but rather an inextricable condition of human existence; by virtue of possessing an individual consciousness, man cannot help but behave purposefully.4

Even doing nothing is an action, for it requires that man consciously refrain from physically intervening in the course of external events; in choosing to remain immobile, man is deliberately substituting the current, unaltered state of affairs for one which would have prevailed had he chosen otherwise. The category of means and ends is thus a necessary presupposition of action, as purposeful behavior implies the existence of a goal or objective whose realization is sought in time and space. In order to act, man must employ some means for the attainment of some end, including his time, knowledge, and physical body. 

Having recognized this category of means and ends, we may then deduce the law of causality from the fact that action would be logically impossible without the expectation that the means employed will successfully realize the ends sought.5 Whether or not this expectation is proven correct ex-post by the outcome of action is irrelevant to the fact that it must necessarily precede all purposeful behavior. Mises explains:

The category means and ends presupposes the category cause and effect. In a world without causality and regularity of phenomena there would be no field for human reasoning and human action. … Where man does not see any causal relation, he cannot act.6

It is in this sense that action may be considered the ultimate foundation of knowledge, for knowledge is structured by the categories of action; not only is knowledge a necessary constituent of every action, but also reflective grasp of the meaning of action is how we come to know what we know. Hence, every category of action is also a category of understanding, and vice versa; to reason and to act are both aspects of one and the same phenomenon.7

Furthermore, it should be noted that action is the process by which the mind interacts with reality. In employing his knowledge and physical body in physical space, man utilizes his reason to intervene in the course of external events. By implication, then, every intuitive and categorical object8 is realized in the external world by action, including but not limited to the laws of logic, the laws of arithmetic, and the laws of geometry9; it is through action that synthetic a priori propositions become factual statements about the structure of reality. 

The insights of praxeology thus provide the solution to bridging the gap between the mental and the real. Unlike transcendental idealism, which conceives of reality as conforming to the principles of cognition, praxeology recognizes that knowledge is structured by the categories of action and thus reflects the operation of reality as much as it does the operation of the mind.10 In categorizing a situation in time and space, the mind immediately affirms the existence of an immanent reality lying outside the realm of thought.11

III. Praxeology and Mathematics

It is here that we may begin to investigate the ontological status of mathematical entities. According to the prevailing doctrine of formalism in the philosophy of mathematics, such disciplines as arithmetic and geometry are reducible to mere games in which meaningless constructs are manipulated with the aid of arbitrary transformation rules; hence, the proposition, 1+1=2, or, The sum of the internal angles of a right triangle is equal to 180 degrees, has no meaning outside of the mathematical schema. Abstract entities such as numbers and shapes are not real objects at all but rather instruments constructed for the purpose of performing symbolic manipulations.

Contra formalism, however, both arithmetic and geometry have as their foundation synthetic a priori judgements whose propositional content is affirmed via action. In his On Praxeology and the Praxeological Foundations of Epistemology (1993), philosopher-economist Hans-Hermann Hoppe writes12:

Arithmetic and its character as an a priori-synthetic intellectual discipline is rooted in our understanding of repetition—the repetition of action. [I]t rests on our understanding the meaning of ‘do this—and do this again, starting from the present result.’13

More particularly, arithmetic may be understood as the categorization of objects in time and space into “constructively identified”14 units and the investigation of the logical relations between them.15 The concept of numbers, then, is not a meaningless construct with only instrumental merit but rather an object whose existence is realized in the external world by and through action; in categorizing spatio-temporal reality into discrete, meaningful units, man cannot help but affirm the real existence of numbers. While the proposition, 1+1=2, may be contradicted by pure observation, as in the case where tying two smaller knots together results in the formation of one larger knot,16 the sum of one constructively identified unit of something with another such unit cannot be equal to anything other than two units of the same kind.

Likewise, geometry is not the arbitrary construction of shapes and patterns but rather the formalization of those principles in accordance with which the mind apprehends spatial relations; in the absence of such principles, neither action nor measurement in time and space would be possible. Hoppe explains:

Action is the employment of a physical body in space. Without acting there could be no knowledge of spatial relations and no measurement. [G]eometry … is no more and no less than the reconstruction of the ideal norms underlying our construction of such homogeneous basic forms as points, lines, planes and distances… It is in this sense that geometry is an a priori science and must simultaneously be regarded as an empirically meaningful discipline because it is not only the very precondition for any empirical spatial description, but it is also the precondition for any active orientation in space.17

IV. The Indispensability Argument

Let us now turn our attention to indispensabilist platonism in the philosophy of mathematics, especially in relation to the analysis of the prime-numbered life cycles of periodical cicadas provided by philosopher of science Alan Baker. The Quine-Putnam indispensability argument for belief in the existence of mathematical entities may be summarized as follows: Because mathematical entities play an indispensable role in our best scientific theories, we ought to believe in their existence. To recognize the validity of such theories without committing to the entities they postulate amounts to upholding a double standard.

As scientific instrumentalists have been too eager to point out, however, this argument is untenable in light of the fact that some of our best scientific theories postulate entities with no correspondence to reality; while such formulations as point masses, perfectly continuous fluids, and frictionless surfaces may be indispensable to analyzing physical phenomena, they do not carry with them any kind of ontological commitment.

An explanatoriness clause has thus been introduced into the original indispensability argument, yielding the following, “enhanced”18 statement: We ought to believe in the existence of mathematical entities that play an indispensable, explanatory role in our best scientific theories. One cannot posit an entity in the explanation of a phenomenon without simultaneously committing to its existence. In his Mathematical Explanation in Science (2009), Baker writes:

The idea, shared by both sides of the debate, is that inserting the word ‘explanatory’ … makes [the indispensability argument] more plausible because it restricts attention to cases where we can posit the existence of a given entity by inference to the best explanation.19 Hence, mathematical entities may be shown to exist if and only if they are explanatory in their own right of some physical phenomenon. 

V. Periodical Cicadas and Prime Numbers

The case study of the prime-numbered life cycles of periodical cicadas presents an interesting example of a phenomenon whose explanation may lie in the mathematical component of a scientific theory. Biologists have observed that cicadas belonging to the various species of the North American genus Magicicada spend much of their lives underground as nymphs before emerging to mate. A notable trait shared by all of these species is the number of years they spend underground, which is either 13 or 17 depending on the species in question.

Moreover, the members of any one species of Magicicada in a given geographical region emerge synchronously, i.e., within the same couple of days, after which they reproduce and die a few weeks later. Two main explanations have been proposed for this peculiar phenomenon, both of which rely on the mathematical concept of primeness: i) Prime-numbered life cycles significantly reduce the likelihood of coinciding with periodically synchronized predators,20 and ii) Prime-numbered life cycles minimize the frequency of interbreeding by ensuring that each species of Magicada mate with another of the same cycle period.21

For example, a cicada with a non-prime-numbered life cycle of 12 years will coincide with predators that emerge once every 1, 2, 3, 4, 6, or 12 years, whereas a cicada with a prime-numbered life cycle of 13 years can only coincide with predators that also emerge once every 13 years22; likewise, a cicada with a non-prime-numbered life cycle of 16 years is likely to mate with other cicadas that emerge once every 1, 2, 4, 8, or 16 years, whereas a cicada with a prime-numbered life cycle of 17 years can only mate with other cicadas that also emerge once every 17 years.

It should be noted that while these explanations also rely on non-mathematical concepts such as biological evolution and natural selection, it is the mathematical concept of primeness, and, more particularly, the principles of number theory, that explain why periodical cicadas have prime-numbered life cycles. As Baker has pointed out,

the purely mathematical component … is both essential to the overall explanation and genuinely explanatory in its own right. In particular, it explains why prime periods are evolutionarily advantageous in this case. Thus … this application of mathematics yields ‘explanatory power.’23

VI. The Problem of Circularity

A common objection to Baker’s argument has been that it begs the question against the mathematical nominalist by assuming the existence of entities with the property, primeness, from the outset. The charge of circularity is leveled against the mathematical platonist by pointing out that periodical cicadas are described to have prime-numbered life cycles even before ontological commitment to numbers has been justified. There is no reason why the mathematical nominalist ought to accept the premise, Periodical cicadas have prime-numbered life cycles, in the absence of a justification for belief in the existence of mathematical entities. In his Inference to the Best Explanation and Mathematical Realism (2008), philosopher of science Sorin Bangu writes:

Baker begs the question against the nominalist. [I]f [the proposition], The number attributed to the … life-cycle period of cicadas measured in years is prime, is taken to be true, this can’t hold unless there is a mathematical object to which the property ‘is prime’ applies. Therefore, by taking the explanandum as being true, Baker assumes realism before he argues for it.24

Baker’s strategy for circumventing this problem has been to temporarily suspend judgment on the primeness of the life-cycle lengths of periodical cicadas and instead advance the broader proposition, “The lengths (in years) of the life cycles of periodical organisms are likely to be prime,”25 as a common explanation for why different species of Magicicada all have prime-numbered life cycles. Given the appropriate physical conditions, then, the observational statement, Periodical cicadas have life-cycle lengths of 13 and 17 years26, may be said to follow as a specific consequence of the hypothesis that prime-numbered life cycles are evolutionarily advantageous to periodical organisms. Hence, the platonist may avoid begging the question against the nominalist by committing to his conclusion only after a common explanation for the prime-numbered life cycles of periodical cicadas has been established.27 Baker explains:

This pattern of argument combines inference to the best explanation with a form of ‘bootstrapping,’ but it does not involve begging the question against the nominalist. The basic data, (11) and (12)28, [are] acceptable to the nominalist. Theses (11*) and (12*)29 are not nominalistically acceptable, but they are initially advanced only tentatively. There are then two possible outcomes of the search for an explanation for (11*) and (12*). Either there is a common explanation of the primeness of the two period lengths or there is not. If there is such an explanation, then this can also be turned into an explanation of (11) and (12), by adding the subspecies-specific ecological constraints.30

This argument, however, fails to address Bangu’s central point, as what is taken to be the common explanation for the prime-numbered life cycles of periodical cicadas still presupposes the existence of abstract, mathematical entities. The proposition, “The lengths (in years) of the life cycles of periodical organisms are likely to be prime,”31 rests on the assumption that there are real objects to which we may ascribe the tendency to exhibit the property, primeness. While introducing an explanation for the premise, Periodical cicadas have prime-numbered life cycles, may give room for the nominalist to infer the existence of numbers from a general hypothesis, the validity of committing to mathematical entities is still assumed from the outset. In employing a form of “bootstrapping,”32 Baker has thus merely postponed the problem to a later part of the argument.

It is here that we encounter the hamartia of the enhanced indispensability argument, namely that one cannot demonstrate the existence of mathematical entities using inference to the best explanation without assuming what is to be proved. For an explanation is nothing more than an answer to a why question.33 In searching for an explanation, we seek to answer the question, Why is it so? Any endeavor to explain a phenomenon must first assume the existence of what is to be explained, i.e., the subject matter of the relevant why question. Hence, one cannot ask the question, Why do periodical cicadas have prime-numbered life cycles?, without implicitly affirming the proposition, Periodical cicadas have prime-numbered life cycles, thereby begging the question against the nominalist.34 The very fact of asking a why question implies belief in that about which it is being asked.

VII. The Praxeological Solution

The solution to this seemingly insuperable dilemma is provided by the insights of praxeology. Once we recognize that the year is a discrete, “constructively founded”35 unit of time in terms of which we have chosen to categorize the life-cycle lengths of periodical cicadas, we cannot help but affirm the ontological correspondence between numbers and reality. The mathematical nominalist may certainly refuse to acknowledge this correspondence, but, in so doing, he would contradict the very principles which allow for the meaningful apprehension of circumstances in time and space.

For in order to describe periodical cicadas as having life-cycle lengths of 13 and 17 years, one must, through the repetition of action, categorize the lifespan of periodical cicadas into 13 and 17 time-units called years. As we have seen, every category of understanding is also a category of action. More particularly, the “structural constraints”36 on knowledge exist within the “categorical framework of means and ends.”37 Since action is the process by which man utilizes his reason to influence the course of external events, every intuitive and categorical object is realized in the external world by and through action.38 In constructing or identifying a certain quantity of discrete, meaningful units, then, man must necessarily presuppose an ontological correspondence between the concept of numbers and the structure of reality. It is the task of arithmetic or number theory to investigate the logical relations between such units and formalize them into a systematic whole.

The nominalist thus cannot deny the general proposition, Numbers exist, or the particular, number-theoretic proposition, Primeness is a real property of 13 and 17, without falling into a performative contradiction. That is, by asserting the counterproposition, Numbers do not exist, he would have already demonstrated that he understands the conceptual content of numbers, and, by implication, what it means to categorize objects in time and space into constructively identified units through the repetition of action; put simply, the mathematical nominalist would refute his own argument by implicitly affirming its negation. Given that primeness is an intrinsic property of numbers whose only factors are 1 and themselves, we must then acknowledge the statement, Primeness is a real property of 13 and 17, as a necessary consequence of the fact that numbers exist as real objects. Hence, we arrive at our conclusion: 13 and 17 are real, prime-numbered mathematical entities.39

VIII. Objections

In the context of the enhanced indispensability argument, the mathematical component of Baker’s analysis of the prime-numbered life cycles of periodical cicadas has been criticized for allegedly relying on an arbitrary choice of units. If the lifespan of periodical cicadas were expressed in time-units other than years such as days, weeks, or lunar months, the number-theoretic concept of primeness would no longer play an explanatory role. For example, 13 and 17 years could instead be expressed as 4,748 and 6,205 days, 678 and 886 weeks, or 161 and 210 lunar months, none of which are prime-numbered periods. The choice of years as time-units is not required by the phenomenon in question, so the primeness of 13 and 17 cannot genuinely explain the cyclical pattern exhibited by periodical cicadas. This objection, however, may be dispelled once the physical significance of the year has been recognized. As Baker has pointed out,

the year as time-unit [is] bound up with genuine physical features of the situation. Years correspond to a physical regularity (one complete orbit of the Sun by the Earth). [T]his regularity is of direct relevance to the life cycles of organisms since it affects local features such as temperature and amount of daylight.40

We are thus empirically justified in our choice of years as time-units.  

A similar charge of arbitrariness has been leveled against the use of numbers in the explanation of the relevant phenomenon. The nominalist contends that acknowledging the explanatory role played by the principles of number theory need not require ontological commitment to numbers, for simply translating the latter into first-order logic would eliminate them altogether. To use Baker’s example, the statement, “The number of F’s is 2,”41 can be restated as, “∃x∃y(Fx ∧ Fy ∧ x ≠ y ∧ ∀z(Fz ⊃ (z = x ∨ z = y))),”42 doing away with the need to quantify over numbers. Hence, one can accept that primeness is a central aspect of the survival strategy employed by periodical cicadas without recognizing the existence of mathematical objects.

This view, however, overlooks the fact that translating quantitative judgements into first-order logic would not rid them of their propositional content. More particularly, the proposition, ∃x∃y(Fx ∧ Fy ∧ x ≠ y ∧ ∀z(Fz ⊃ (z = x ∨ z = y))), is not a set of meaningless symbols but rather a statement whose content is conveyed by logical variables, quantifiers, connectives, and identities; the expression, There is some x and y such that x is F, and y is F, and … , is merely a paraphrase of the statement, The number of F’s is 2, whose meaning has not been lost in the translation. Moreover, verbal reasoning is the very basis of formal logic, for without meaningful words and sentences, such symbols as ∧, ⊃, and =,43 would have no conceptual referents. As philosopher-economist Murray Rothbard has pointed out in a similar discussion of the use of symbols in mathematical logic,

[F]ormal logic deals with the necessary and fundamental laws of thought, which must be verbally expressed, and logistics is only a symbolic system that uses this … verbal logic as its foundation.44

The proposition, The number of F’s is 2, is thus an inextricable correlate of the statement, ∃x∃y(Fx ∧ Fy ∧ x ≠ y ∧ ∀z(Fz ⊃ (z = x ∨ z = y))).

In addition, as we have already seen, primeness is a number-theoretic concept which denotes a property of numbers that are indivisible by factors other than 1 and themselves. In order to speak of primeness, one must also refer to the objects which possess the designated characteristic. Primeness is not primeness as such but rather primeness of something, that which has the property of being prime. Just as the concept of extension has meaning only in relation to extended things, so the notion of primeness involves reference to prime numbers. Consequently, the nominalist who concedes that primeness figures in the explanation for the life-cycle lengths of periodical cicadas must necessarily acknowledge the existence of numbers, and, by implication, mathematical entities, on pain of falling into a contradiction. 

IX. Conclusion

The foregoing discussion of praxeology and its place in the philosophy of mathematics has demonstrated that mathematical entities are realized in the external world by human action. In categorizing objects in time and space into units constructed through the repetition of action, the mind immediately affirms the ontological correspondence between numbers and reality. On this account, we have observed the logical incoherence of mathematical nominalism in the performative contradiction entailed by the attempt to deny the existence of numbers; in asserting the proposition, Numbers do not exist, the nominalist has already demonstrated an understanding of what it means to categorize spatio-temporal objects into units founded on the logical structure of action.

Furthermore, the Quine-Putnam indispensability argument has been shown to fail as a justificatory basis for mathematical realism on two counts: it cannot explain the lack of ontological commitment to such idealized entities as point masses and perfectly continuous fluids, and the introduction of an explanatoriness clause merely begs the question against the nominalist. Belief in the existence of mathematical entities, then, is justified not by their indispensability to some of our best scientific theories but rather by their relation to the categorical framework of action; it should be no surprise that mathematical entities, and, more particularly, numbers, are indispensable to scientific analysis, for they constitute a basic structural aspect of the external world.45

  1. Clarifying the precise meaning of “best” in this context is outside the scope of this paper.
  2. Baker, Alan. “Mathematical Explanation in Science.” The British Journal for the Philosophy of Science 60, no. 3 (2009): 611–33. https://doi.org/10.1093/bjps/axp025, p. 613 (emphasis in original).
  3. As formulated by philosopher-economist Ludwig von Mises.
  4. This, of course, does not imply that all human behavior is purposeful; such involuntary movements as breathing and digestion must be differentiated from voluntary action.
  5. Unlike in mathematics, the process of logical deduction used in praxeology is material rather than formal, for the content of each relevant proposition must be known in order to arrive at its necessary implications.
  6. Mises, Ludwig Von, and Bettina B. Greaves. Human Action: A Treatise on Economics (Indianapolis: Liberty Fund, Incorporated, 2014), p. 22 (emphasis in original).
  7. See Mises, The Ultimate Foundation of Economic Science, p. 42, as cited in Hoppe, The Economics and Ethics of Private Property, p. 279. Mises writes: “Both, a priori thinking and reasoning on the one hand and human action on the other, are manifestations of the mind. Reason and action are congeneric and homogeneous, two aspects of the same phenomenon” (my emphasis).
  8. Every law or principle that may be derived from the pure intuitions of time and space and the categories of understanding.
  9. On the laws of logic as synthetic a priori principles, see Hoppe, The Economics and Ethics of Private Property, pp. 284-5; a more detailed discussion of arithmetic and geometry as synthetic a priori disciplines will follow below.
  10. See Hoppe, The Economics and Ethics of Private Property, pp. 282-3. Hoppe writes: “Understood as constrained by action categories, the seemingly unbridgeable gulf between the mental on the one hand and the real, outside physical world is bridged. So constrained, a priori knowledge must be as much a mental thing as a reflection of the structure of reality since it is only through actions that the mind makes contact with reality…” (my emphasis).
  11. This reality is not mind-independent, for it exists only insofar as there is an acting subject. The postulates of praxeology, however, really do exist out there, for action occurs in time and in space within the world.
  12. In reference to the constructivist method of philosopher-mathematician Paul Lorenzen and his Erlangen School.
  13. Ibid., p. 286 (my emphasis).
  14. Ibid., p. 286 (my emphasis).
  15. Likewise, in his History of Philosophy (1941), philosopher Julián Marías writes: “[Synthetic a priori] judgements are possible in mathematics, a science that is based on constructions of concepts. The validity of mathematics is based on the a priori intuition of relationships between spatial figures and numbers which are based on the temporal succession of units” (p. 290, emphasis in original).
  16. This would seem to indicate a scenario in which the expression, 1+1, has a value of 1 rather than 2.
  17. Hoppe, Hans-Hermann. The Economics and Ethics of Private Property: Studies in Political Economy and Philosophy. Auburn, AL: Ludwig von Mises Institute, 2006, p. 288 (my emphasis).
  18. Baker, Alan. “Mathematical Explanation in Science.” The British Journal for the Philosophy of Science 60, no. 3 (2009): 611–33. https://doi.org/10.1093/bjps/axp025, p. 613.
  19. Ibid., p. 613 (my emphasis).
  20. See Goles, Schulz, and Markus, Prime Number Selection of Cycles in a Predator-Prey Model, p. 33, as cited in Baker, Are There Genuine Mathematical Explanations of Physical Phenomena?, p. 231.
  21. For a more extensive treatment of this case study, see Baker, Are There Genuine Mathematical Explanations of Physical Phenomena?, pp. 229-33.
  22. Ibid., p. 231.
  23. Ibid., p. 233 (emphasis in original).
  24. Bangu, Sorin Ioan. “Inference to the Best Explanation and Mathematical Realism.” Synthese 160, no. 1 (2006): 13–20. https://doi.org/10.1007/s11229-006-9070-8, p. 18 (my emphasis).
  25. Baker, Alan. “Mathematical Explanation in Science.” The British Journal for the Philosophy of Science 60, no. 3 (2009): 611–33. https://doi.org/10.1093/bjps/axp025, p. 621.
  26. Which the nominalist may accept due to the lack of any reference to primeness.
  27. For a more detailed exposition of this argument, see Baker, Mathematical Explanation in Science, pp. 619-22.
  28. “(11) The length (in years) of the life cycle of cicada subspecies A is 13, (12) The length (in years) of the life cycle of cicada subspecies B is 17” (Ibid., p. 260, my emphasis).
  29. “(11*) The length (in years) of the life cycle of cicada subspecies A is prime, (12*) The length (in years) of the life cycle of cicada subspecies B is prime” (Ibid., p. 260, my emphasis).
  30. Ibid., p. 621 (my emphasis).
  31. Ibid., p. 621.
  32. Ibid., p. 621.
  33. C., Van Fraassen Bas. The Scientific Image. Oxford: Clarendon Press, 1980, p. 134.
  34. Even if this question were rephrased as, Why do periodical cicadas have life-cycle lengths of 13 and 17 years?, in order to better reflect Baker’s “bootstrapping” strategy, the charge of circularity would still hold given the presence of numbers in the relevant why question. Translating 13 and 17 into first-order logic will not eliminate this problem, as the content of the resulting statements would be no different from that of the original quantities; more will be said about this in a later discussion.
  35. Hoppe, Hans-Hermann. The Economics and Ethics of Private Property: Studies in Political Economy and Philosophy. Auburn, AL: Ludwig von Mises Institute, 2006, p. 286 (my emphasis).
  36. Ibid., p. 281 (my emphasis).
  37. Ibid., p. 292.
  38. See the above discussion of praxeology and arithmetic.
  39. Whereas Baker’s enhanced indispensability argument attempts to deduce the existence of numbers from a set of empirical premises, the praxeological basis for mathematical realism identifies the necessary preconditions for reasoning about numbers in the first place.
  40. Baker, Alan. “Mathematical Explanation in Science.” The British Journal for the Philosophy of Science 60, no. 3 (2009): 611–33. https://doi.org/10.1093/bjps/axp025, p. 615 (my emphasis).
  41. Ibid., p. 619.
  42. Ibid., p. 619.
  43. Which denote the connectives and, if-then, and is.
  44. Rothbard, Murray Newton. Man, Economy, and State a Treatise on Economic Principles. Auburn, Ala: Ludwig von Mises Institute, 2009, p. 76 (my emphasis).
  45. The Quine-Putnam indispensability argument is thus a mere implication of the praxeological justification for belief in the existence of mathematical entities.