(Montessori Schools, 2018)
TABLE OF CONTENTS
1. Introduction: From Mind to Mathematics
2. The Nature of Mathematical Entities: Nothing Works
3. Set Theory: Should One Believe?
5. Mathematical Fictionalism
6. Mathematical Realism
7. Hanna’s Neo-Intuitionism as a Way Out of the Impasse?
8. Conclusion: From Mathematics to Mind
This essay will be published in four installments, two sections per installment; this is the third, and contains sections 5 and 6.
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5. Mathematical Fictionalism
Let us begin with a consideration of fictionalism. The idea that mathematical objects do not exist, so that sentences like “1 + 1 = 2” are false, or at least not true, flies in the face of the way mathematicians view their subject matter. It also is inconsistent with psychological research indicating that there is a biological origin to at least arithmetic, an expression of the “deep structure’” of human perception (Grace et al., 2023, 2024). Thus, one attempt to deal with this objection is Meinongianism, the view that mathematical theories produce true descriptions, but these objects do not exist, but there are still true mathematical statements. In more recent times, this position has been defended by, as he was then known, Richard Routley (Routley, 1980) and Graham Priest (Priest, 1983, 2003, 2005). The position holds that even sentences involving contradictory objects can be true. Thus, “the round square, is round and square,” is true, but “the round square is triangular,” is false. Balaguer objected to the Meinongian account, that “Fa” can be true, even if there is no object denoted by “a,” since in the standard account of truth, if there is no object, then the sentences “Fa” is not true (Balaguer, 2009: p. 49). But this begs the question against Meinongianism.
Routley, in Exploring Meinong’s Jungle, defends noneism, that everything is an object of discourse, but abstract entities do not exist. He rejects the “reference theory,” the ontological assumption that all proper subject terms of true statements must have an actual reference, that is that the truth of Fx implies that x exists (Routley, 1980: p. 24). Routley’s Independence Thesis is that “items can and do have definite properties even though nonentities” (Routley, 1980: p. 28). Being is not part of the characterization of an object; unlike in existentialism, essence precedes existence. But what is, or was existence for Routley? He does not give a definition of “exists,” yet does say that there is a stable meaning of the word. However:
[A]s the Platonist hastens to point out, there is nothing to prevent us using the word “exists” in any way we please. However, it is not unreasonable to require that before we adopt a usage which is misleading and liable to cause dislocation and equivocation, it should be clear that the gains for doing so at least outweigh the losses. (Routley, 1980: p. 634)
Routley discusses the meaning of existence in chapter 9 of Exploring Meinong’s Jungle and Beyond. He shows at length that all of the standard accounts are unsatisfactory, often being circular, which is to be expected with such a fundamental concept (Routley, 1980: p. 701) He concludes that “No rock-hard criterion for what exists has appeared” (Routley, 1980: p. 730). This makes it philosophically problematic to then claim that abstract objects, such as sets do not exist, as Platonism supposes (Routley, 1980: p. 730).
Further, Routley did not produce an account of mathematical truth which was coherent with his Meinongianism. Indeed, he rightly notes that there is a circularity problem with the truth definition, “Statement A is true in selected framework M”:
[I]n order that the definition of “statement A is true in selected framework M” shall provide us with a definition of truth, of “A (of L) is true” is that the framework M selected is the correct one, one whose base world T is the factual world, and represents the class of true statements. Thus, in order to define truth, we have already, in effect, to be supplied with the class of truths, T. An irremediable circularity thus appears to have crept into the business of giving a semantical definition of truth. (Routley, 1980: p. 330)
Routley then concludes that each theory of truth can furnish its own criterion for determining M; e.g., for the coherence theory, M is the model that coheres with experience (Routley, 1980: p. 331). And, “every theory is correct according to its own lights” (Routley, p. 334). The corresponding pluralism adopted by Routley, when he changed his name to Richard Sylvan, in Transcendental Metaphysics (Sylvan, 1997), means that he has no reason to convince Platonists to abandon their position, and no independent reason for accepting Meinongianism.
Priest gave an alleged anti-realist account of mathematical truth (Priest, 1983), which we will now consider. We will not discuss his own version of Meinongianism, modal noneism (Berto, 2014), since even if this position does give a satisfactory account of the logic of non-existence and objects, it still does not show by its “existence” that, say, mathematical Platonism is false, because there could be independent arguments for that position quite apart from referential considerations, such as the practice of working mathematicians via their usage, and possible indispensability (Baker, 2001).
Priest set out to give the truth conditions for mathematical statements, without the use of abstract entities. He illustrated this approach with a discussion of both arithmetic and set theory, but we will consider arithmetic only (Priest, 1983: pp. 50-52). Priest first defines the language of arithmetic, the signs and terms of the language. There is one constant 0, a one-place function S, and the two-place functions + and x. Terms are formed recursively from 0 using S, +, and x. A numeral is any term with a string of S’s preceding a 0. Atomic formulas are of the form “t1 = t2 ,” where t1 and t2 are terms. The absurdity symbol is f. The set of formulas F is the closure of the set of atomic formulas, with “the standard truth conditions for first-order languages with the implication operator and substitutional quantifiers” (Priest, 1983, 51). The truth conditions for atomic sentences are then given using canonical forms:
The canonical form t*of a term t is a numeral which can be given a recursive definition as:
0* = 0
(St) = St*
(t1 + t2)* = the term obtained by prefixing all the S’s at the beginning of t1*to t2*
(t1 x t2)* = the term obtained by replacing every occurrence of S at the beginning of t2* by as many S’s as commence t1*.
The truth condition of ‘t1 = t2’is:
‘t1 = t2’ is true iff t1* is the same ast2*.
That is all fine, but what seems to be missing is an account of the truth of the atomic sentences to get the show going. Priest writes “[f]irst let us suppose that we have an account of the truth conditions of atomic sentences” (Priest, 1983: p. 51). However, let us not suppose this, and ask instead: how are atomic sentences of arithmetic shown to be true? One suggestion is to proceed inductively showing that sentences such as “1 + 1 = 2” is true, “1 + 2 = 3” is true, “1 + 2 = 4” is false and so on (Priest, 1983: p. 52). But that will require an infinite number of proofs, which cannot be done. Priest then suggests taking say the Peano axiom system, and showing that all its theorems are true under his proposed truth conditions (Priest, 1983: p. 52). But that too will involve an infinite number of proofs, which will not work. Finally, Priest suggests what he thinks is the better approach:
The realist has a way of specifying precisely these sentences of arithmetic which are true, viz., those sentences which hold in (that mathematical fiction) the “standard model.” Now it is easy to prove by simple induction over formation that a sentence of arithmetic is true under the truth conditions I have given iff it holds in the “standard model.” Thus, we can argue ad hominem against the realist that the truth conditions are right. If his account fits the pretheoretic data, so does ours. (Priest, 1983: p. 52)
However, for our purposes, which is to conduct a neutral evaluation of all these positions, the ad hominem argument will not be convincing, since the truth of mathematical realism is as much under critical scrutiny as Priest’s alternative. We thus do not ascertain in any non-question-begging fashion, the method of proving the truth of the atomic sentences of arithmetic.
Feng Ye (Ye, 2010, a), has proposed that physicalism about cognitive subjects implies mathematical nominalism. If physicalism is true, there will be a complete physical description of everything in the world consisting of a mathematician doing mathematics. In proving a statement, “there are an infinite number of primes,”
[w]e won’t say anything like “our brains are committed to the existence of numbers.” Our acceptance of the sentence is also a physical event involving some neural activities in our brains and a neural circuitry (that associated with the sentence interacting with other physical things. (Ye, 2010a: p. 134)
As we see it, however, even if physicalism is true, it does not imply mathematical nominalism. There is no demonstration that abstract mathematical entities do not exist merely because physicalism about the world, including the mind, is supposedly true. And, if there were the implication Physicalism ® Nominalism, then it could be shown that nominalism was unacceptable for some independent reasons, then we could then reject physicalism, which is incorrect, the position could not be refuted in that way. There could be good philosophical reasons for physicalism about the world, but also reasons against nominals, in fact in another paper (Ye, 2010, b), Ye says that mathematical anti-realism must in general explain: (1) what, if anything exists in mathematics; (2) the objectivity of mathematics; (3) the prima facie apriority and necessity, and (4) how mathematics is applicable to reality (Ye, 2010,b, 15). And he concludes: “No current anti-realistic philosophies can meet all these challenges and constraints (Ye, 2010, b, 15). We accept that conclusion and move now to consider mathematical realism.
6. Mathematical Realism
The main problem with mathematical realism is that if mathematical entities do not exist in space-time, then there is no causal interaction between these entities and the human mind, which is the famous Benacerraf objection (Benacerraf, 1973). He presupposed a causal theory of knowledge, but the same point can be made without that, which is that for Platonism, there is a difficulty, indeed a mystery as to how reference, and understanding of mathematical entities is possible, as Platonic entities are outside of space-time, but the human mind is not (Hodes, 1984). Here the appeal is usually made to mathematical intuition, which somehow without contact generates knowledge. But that rational faculty is seen by critics as “mysterious.” Now, the point of this paper is to recognise just that, that the queen of the sciences, has mystery at its heart, so given that, we should not be disturbed when we find the mind as a mystery as well.
Balaguer has argued that a Full Blooded Platonism (FBP), where “all the mathematical objects that possibly could exist actually do exist” (Balaguer, 2009: p. 59) can solve the epistemological problem noted without any sort of information-transferring contact, between human minds and abstract objects:
Since FBP says that all the mathematical objects that possibly could exist actually do exist, it follows that if FBP is correct, then all consistent purely mathematical theories truly describe some collection of abstract mathematical objects. Thus, to acquire knowledge of mathematical objects, all we need to do is acquire knowledge that some purely mathematical theory is consistent. (It doesn’t matter how we come up with the theory; some creative mathematician might simply “dream it up.”). But knowledge of the consistency of a mathematical theory—or any kind of theory, for that matter—does not require any sort of contact with, or access to, the objects that the theory is about. Thus, the Benacerrafian lack-of-access problem has been solved; we can acquire knowledge of abstract mathematical objects without the aid of any sort of information-transferring contact with such objects. (Balaguer, 2009: p. 59)
One might first ask how paraconsistent mathematics, based upon the study of inconsistent mathematical objects (Mortensen, 1995) fits into this picture? But leaving that point aside, the problem with the appeal to consistency, that mathematical knowledge requires knowing that some mathematical theory is consistent, runs into the problem of Gödel’s Second Theorem: that for any consistent system S, where elementary arithmetic can be carried out, the consistency of S cannot be proven in S. This result applies to all systems upon which ordinary mathematics is based, such as ZFC set theory. So, while there may be consistency proofs of some mathematical theories, the consistency of the whole of mathematics is not provable. Even regardless of Gödel’s Second theorem, if there was an alleged proof of the consistency of all of mathematics, MM, then one could in turn request a proof that MM is consistent. Appealing to MM itself is circular, and if another method is used MM1, then an infinite regress is generated (Lakatos, 1962). And finally, if Brouwer is right (Brouwer, 1967), then classical mathematics is inconsistent, which if true, would undermine Balaguer’s argument among other things. As well, there is inconsistent mathematics (Mortensen, 1995).
Balaguer has also argued that for FBP and anti-Platonism in the form of fictionalism, there is no good reason for choosing one over the other; there is no fact of the matter about which position is correct (Balaguer, 2009: p. 90). His argument is that first FBP and fictionalism are in agreement on most aspects of their philosophy except the existence of abstract objects. But this disagreement cannot be settled either directly or indirectly. It cannot be settled directly as there is no direct access to the mathematical realm, so there is no way of knowing hat such entities exist. The indirect route of examining the consequences of the positions also fails, as the only material difference is over the question of the existence of abstract entities. And as well, there is no fact of the matter about the existence of mathematical objects existing outside of spacetime. We do not know what a possible world would be like where objects did exist outside of spacetime. If so, there is no fact of the matter as to what possible world counts as there being such objects exist outside of spacetime. And if there is no fact of the matter as to which possible worlds count as to where there exist abstract objects outside of spacetime, then there is no fact of the matter as to the actual world being such a world (Balaguer, 2009: p. 95).
If these arguments are accepted, then we would be left with the situation that the question of the existence of abstract objects outside of spacetime is unsolvable. As he says, that may be thought to “undermine the two views” (Balaguer, 2009: p. 35), which seems a reasonable conclusion. And as well, this would support the position stated at the beginning of his paper, that the philosophy of mathematics leads to metaphysical mystery. We are quite prepared to accept this, holding to the position of epistemic humility, that there are limits to human knowledge, and hence unsolvable mysteries that we just live with a brute fact. But there may be an alternative.
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