Mark Balaguer and Platonism: Part 1

Page last updated 21 Jan 2023

In an article Mathematical Platonism (Footnote: Mathematical Platonism by Mark Balaguer: a chapter in the book Proof and other dilemmas: mathematics and philosophy (Vol. 59. MAA, 2008) editors Bonnie Gold and Roger Simons.) Mark Balaguer summarizes the essence of the content of his book Platonism and anti-Platonism in mathematics (Footnote: Platonism and anti-Platonism in mathematics by Mark Balaguer, Oxford University Press on Demand, 1998.) where he claims that he gives an unbiased overview of the arguments for and against Platonism.

You can also see an article by Balaguer at Stanford Encyclopedia: Fictionalism in the Philosophy of Mathematics which covers the material in a similar manner.

His conclusion is the same in both; in the book, he writes: (Footnote: From Balaguer’s book Platonism and anti-Platonism in mathematics:
“My official view, then, is distinct from both FBP [full blooded Platonism] and fictionalism. I endorse the FBP-fictionalist interpretation, or picture, of mathematical theory and practice, but I do not agree with either of the metaphysical views here. More precisely, I am in agreement with almost everything that FBP-ists and fictionalists say about mathematical theory and practice, but I do not claim with FBP-ists that there exist mathematical objects (or that our mathematical theories are true), and I do not claim with fictionalists that there do not exist mathematical objects (or that our mathematical theories are not true).” )  (Footnote: From Balaguer’s article Mathematical Platonism:
“That is, there are no good arguments for the claim that there are abstract objects and hence that our mathematical theories are literally true, and there are no good arguments for the claim that there are no abstract objects and hence that our mathematical theories are strictly speaking untrue. Indeed, I argue that for a variety of reasons (most notably because we cannot obtain any information about whether there are any mathematical objects), we could never have any good reason to endorse FBP [full blooded Platonism] over fictionalism or vice versa.” )

Having made the claim that there is only one good argumentative position against Platonism, Balaguer’s principal argument against it is his claim that its proponents cannot accept that: (Footnote: Balaguer uses the term “fictionalists” to refer to anti-Platonists that he considers to be the only anti-Platonists that might be considered to have any sort of a valid argument. In his article PDF Fictionalism, theft, and the story of mathematics (Philosophia Mathematica 17.2 (2009): 131-162) Balaguer expands on his ideas regarding fictionalism.)

“The sentences of our mathematical theories - sentences like ‘3 is prime’ are true.”

with his concomitant implication that the only anti-Platonists with a viable argument must therefore claim that the sentences of mathematical theories are either “false”, or are neither “true” nor “false” - that they are what Balaguer calls “fictions”.

But in doing so, Balaguer, despite his ardent claim to the contrary (see below) is thereby confirming that he is in fact a closet Platonist since his entire argument against anti-Platonists rests on this crucial claim - a claim that necessarily assumes that there is a difference between a sentence of a mathematical theory being “true” and being provable in that theory. By doing so, he assumes that there “exists” a truth value of the sentences of a mathematical system that has always existed independently of the axioms and rules of that system and which is not a physical entity. That is a Platonist notion. (Footnote: Balaguer also makes the error of assuming that a mathematics formulated by Platonists must necessarily be precisely the same as a mathematics formulated by people with no Platonist beliefs, whereas this is patently incorrect, and can be shown to be incorrect, since contradictions arise directly from Platonist beliefs that infinitely many independently mathematical things can simultaneously ‘exist’. See The Courant & Robbins Contradiction, Lebesgue Measure, Sums of infinitely many fractions: 1, Sums of infinitely many fractions: 2, Platonist assumptions of ‘existence’ in the diagonal proof, Cardinal numbers and Why do people believe weird things? )

And one can note that in an earlier paper A Platonist Epistemology (Footnote: Mark Balaguer, “A Platonist Epistemology”, Synthese 103.3 (1995): 303-325, doi:10.1007/bf01089731.) Balaguer makes a strong defence of Platonist philosophy:

“My argument is based upon the adoption of a particular version of platonism - full-blooded platonism - which asserts that any mathematical object which possibly could exist actually does exist.”

which makes Balaguer’s claim of his agnosticism about Platonism rather questionable. In any case it is meaningless to talk about the ‘existence’ of something being “possible” where there can never be any evidence to confirm that its ‘existence’ is possible - see the Appendix below: Full-blooded Platonism.

In any case, a non-Platonist can simply say that some propositions of a mathematical theory can be proved, and the negation of some propositions of that theory can be proved, and it may be the case that for some propositions, neither they nor their negation can be proved, and that he has no need for the faith-based Platonist notion that the “truth” of the propositions of that theory ‘exist’ independently of the rules and axioms of that theory.

One might claim that we can see that a mathematical statement such as “17 is prime” can be shown to be “true” by taking a collection of 17 objects and repeatedly removing exactly the same quantity of objects from that collection, and observing that, unless one removes one object each time, or 17 objects each time, then there will always be some objects left over. And one can repeat this scientific experiment many times and obtain the same result. Indeed, it was from observances of such physical situations that gave rise to the origin of mathematics. It is not surprising that mathematical systems were developed with rules and axioms that give rise to mathematical expressions that can be useful to describe physical situations, so that a mathematical expression such as “17 is prime” is provable within that mathematical system. And so it is not surprising that there is a close connection between some mathematical statements and some physical situations. And one can say that in many cases, mathematics can indeed be a useful tool to describe a physical situation.

But it is a mistake to conflate the scientific statement regarding a physical situation, such as:

“Given a collection of 17 objects, the repeated removal of exactly the same quantity of objects (more than 1 and fewer than 17) will always result in some objects remaining”

with the purely mathematical expression:

“17 is prime”

which does not reference any physical objects.

Where is the implication in a repeated physical experiment of the independent non-physical ‘existence’ of an object that corresponds to a name such as “17” that a human puts on a collection of objects? There simply isn’t any logical implication to that effect, in the same way that there is no logical implication that there is a non-physical existing object that corresponds to the name “red” that a human can assign to certain wavelengths of light.

As pointed out in The Myths of Platonism, the connection between a statement of a mathematical theory and some physical situation can only be established by some initial ostensive interaction between the statements by a human (or machine) and some physical object. Balaguer’s assumption that there must always have “existed” some sort of inherent independent connection - that must have “existed” prior to any human mathematics or any human language, and which is the “truth” value of mathematical statements - is just a demonstration of his underlying Platonist ethos.

In summary, Balaguer’s claim - that the best argument that anti-Platonists can muster is that mathematics is a “fiction” - can be seen to be nothing more than a thinly disguised straw man that should fool no-one. In the same way that there are no good arguments for the non-evidential faith based notion of a god, there are no good arguments for the non-evidential faith based notion of independently existing non-physical mathematical objects, which is why believers have to resort to such straw man pleadings. For more, see Numbers, chairs and unicorns, Platonism, Platonism’s Logical Blunder, Moderate Platonism, The Indispensability Argument and Descartes’ Platonism.

Balaguer’s fallacious claims about statements and objects

In his article, Balaguer claims that anti-Platonists with valid arguments do not dispute that:

“… the claim here is that ‘3 is prime’ has the same logical form as, e.g. ‘Mars is red.’ Both sentences just make straightforward claims about the nature of certain objects. The one makes a claim about the nature of Mars, and the other makes a claim about the nature of the number 3. … if we read ‘3 is prime’ as making a straightforward claim about the nature of the number 3, and if we allow that this sentence is literally true, then we are committed to believing in the existence of the number 3.”

This is egregious nonsense. Only a fool would not dispute it. The expression “Mars is red” does not of itself make a statement about anything. It is only within the context of the English language that it has the interpretation that we allow for it in English. (Footnote: Consider an alien who has no knowledge whatsoever of human culture and is shown the sequence of symbols “Mars is red” - he will have no idea of what that connotates, because he has no knowledge of the required context.) And for a statement such as “Mars is red”, we know that the context is such that Mars has previously being defined as referring to a physical object for which there is scientific evidence. On the other hand, when one states “3 is prime”, one is making a statement about a concept that has previously been defined, and that is no different to stating “Unicorns have horns”, where again one is making a statement about a concept that has previously been defined. In neither of those two cases is there any implication that the defined concept has any real ‘existence’ independently of the definition, nor that the writing down of the expression somehow confirms the ‘existence’ of what it refers to. Both the number 3 and the concept of a unicorn can be defined without assuming that they somehow have an ‘existence’ independent of that definition.

Balaguer’s claim that both sentences “3 is prime” and “Mars is red” make assertions about the nature of existing objects is just another worthless straw man that will only fool the simple-minded.

Conclusion

While Balaguer claims to be attempting an objective overview of Platonism, it is clear that his own underlying Platonist ethos renders his conclusions completely worthless, being little more than Platonist apologist pleading. He has since written more material in the same vein, with the same underlying Platonist ethos, and I discuss this material in Mark Balaguer and Platonism: Part 2.

“Full-blooded Platonism”

The notion of full-blooded Platonism that claims that any mathematical object which possibly could ‘exist’ actually does ‘exist’ is meaningless verbiage. To state as Balaguer does, that the ‘existence’ of a non-physical thing is possible, is simply another way of saying that one can never have information that would confirm the ‘existence’ of the thing. This means that a claim that any such possible thing must ‘exist’ is simply a claim that every postulated non-physical thing must ‘exist’, which is absurd.

 

Such a claim assumes that the simultaneous ‘existence’ of such “possible” things can never result in a contradiction. An example of contradictory possible non-physical things is given by the following postulates:

• It is possible that Binklebots exist; Binklebots are non-physical things that can prove by rigorous logic that Grinklebots cannot ‘exist’.
• It is possible that Grinklebots exist; Grinklebots are non-physical things that can prove by rigorous logic that Binklebots cannot ‘exist’.

By Balaguer’s notion of ‘possible existence’, there is no evidence that Binklebots do not ‘exist’, then it is possible that they ‘exist’, and so they must ‘exist’. Similarly, Grinklebots must ‘exist’. But this results in a contradiction, hence both postulates cannot be reflecting reality. Balaguer’s assumption that anything that can possibly ‘exist’ does ‘exist’ lacks any logical basis. (Footnote: Elsewhere Balaguer claims that full-bodied Platonism “is the only tenable version of platonism” which appears to indicate that his defence of Platonism is rather lacking in substance (PDF A fictionalist account of the indispensable applications of mathematics, Philosophical Studies 83.3 (1996): 291-314).)

 

In the same way, Platonist assumptions of the ‘existence’ of the sums of infinitely many fractions results in similar contradictions, see for example Sums of infinitely many fractions: 1.

Footnotes: