Mark Balaguer and Platonism: Part 2
Page last updated 06 Feb 2023
I discussed some of Mark Balaguer’s material on Platonism on the page Mark Balaguer and Platonism: Part 1. Balaguer has recently published some more material in a similar vein. Looking through the collection of material, it is hard to know how Balaguer can seem so totally oblivious to the fact that his material quite clearly demonstrates an underlying Platonist ethos.
Let’s look first at Mark Balaguer’s 2017 article “Mathematical Pluralism and Platonism” (Footnote: Mark Balaguer, “Mathematical pluralism and platonism”, Journal of Indian Council of Philosophical Research 34.2, 2017, pp.379-398.)
where he discusses the notion that what he calls ‘mathematical pluralism’ is ‘true’, where this ‘mathematical pluralism’ is based on the notion that there is not just one mathematical ‘truth’, but many different mathematical ‘truths’ - that there can be different theories that seem to be competitors, or incompatible - but both can be true. He claims that there is a ‘surprising result’ that “mathematical Platonism and mathematical pluralism are perfectly compatible with one another.”
But the entire notion is based on the Platonist assumption that the mathematical ‘truth’ of a mathematical statement ‘exists’ and is completely independent of any mathematical system. On the other hand, one can state that the provability of that statement is completely dependent on the rules and axioms of a mathematical system, without attaching any unwanted Platonist baggage. Without the Platonist assumption, Mark Balaguer’s statement amounts to nothing more than observing that a statement might be provable in one mathematical system but not in another, or that the negation of that statement might be provable in a different mathematical system. Of course, we might observe that one mathematical system might seem to be better suited to real world applications than another - but that would be subject to the proviso that one could never be absolutely sure that that system would always give results in real world applications that are in accordance with actual real world observations - in the application of a mathematical system to science, the combined system of the mathematics and the scientific application becomes subject to Karl Poppers notion of falsifiability. In short, we can never know with absolute certainty that a given mathematical system is the best system to be used in scientific theories, though we might be able to say that certain mathematical systems appear to be better than others for the application to real world scenarios. (Footnote: See also Is Mathematics Unreasonably Effective?)
Balaguer then spends several pages trying to convince non-Platonists that there might be some way to counter the problematic observation that, for a Platonist, given two contradictory postulated mathematical systems, at least one of them must be ‘false’. We can use Balaguer’s example of the Continuum Hypothesis; for a Platonist, it must either be ‘true’ or ‘false’, that is, for a Platonist, either there ‘exists’ at least one set with a cardinality between the cardinality of natural numbers and that of the real numbers or there is not. Mark Balaguer’s attempt at a fudge by verbose playing with words cannot somehow circumvent this problem.
Balaguer then rehashes his tired old claim, as observed in Mark Balaguer and Platonism: Part 1, that a non-Platonist, because he does not believe in the existence of things for which there is no evidential nor logical basis, is nevertheless, according to Balaguer, obliged to believe certain other things prescribed by Balaguer, who claims that if one rejects Platonism then one also must claim that:
- our mathematical theories are not true,
- our mathematical theories are about objects that do not exist,
- mathematical objects do not exist, but nevertheless our mathematical theories are true.
Both (a) and (c) can be rejected summarily, since, as noted above, the notion of mathematically ‘true’ is a Platonist notion. That leaves a consideration of (b), which can also be summarily dismissed, on the grounds that since an object is a real thing that exists, then the claim is nothing but a meaningless play on words. The notion that a non-Platonist must choose one of the three options is nonsense. A non-Platonist can simply remark that he has no reason to believe in the existence of things for which there is no evidential nor logical basis, and he simply uses certain mathematical definitions that satisfy certain statements of a defined mathematical system, and such definitions do not in any way imply the independent real non-physical existence of anything.
In short, it is clear that Balaguer is still pushing the ridiculous notion that a non-Platonist is somehow obliged to do more than simply reject any belief in the existence of things for which there is no evidential nor logical basis. But a Platonist is no more obliged to do so than to provide any rationale for his non-belief in unicorns other than the fact that there is no evidential nor logical basis to believe that unicorns exist.
Moving onto Balaguer’s 2023 book Mathematical Anti-realism And Modal Nothingism, (Footnote: Mark Balaguer, Mathematical Anti-Realism and Modal Nothingism, Cambridge University Press, 2023.) in section 2.2 Mathematical Realism, he continues to push the absurd notion that there is some sort of argumentative substance in the claim that a statement such as “3 is a prime number” is somehow, of itself, indicative of the independent existence of 3. It essentially amounts to saying, “ I would like the statement ‘3 is prime’ to imply that 3 is an object that exists independently of any mention of 3, therefore 3 is an object that exists independently of any mention of 3.” I had already dealt with that absurd notion in my Part 1 at Balaguer’s fallacious claims about statements and objects.
In his section 2.3 Three Versions of Mathematical Anti Realism, Balaguer claims that there are just three valid stances of non-Platonism:
- Paraphrase Nominalism and the Rejection of [the premise that the statement “3 is prime” implies that 3 is an object].
- Mathematical Error Theory and the Rejection of [the premise that mathematical sentences and theories provide true descriptions of abstract objects]
- Deflationary Truth Nominalism and the Rejection of [the premise that mathematical sentences and theories provide true descriptions of mental objects, presumably ideas in our heads]
For a non-Platonist, there’s no difficulty with (i) and (ii), one is simply rejecting notions for which there is no evidential nor logical basis. As for (iii), the non-Platonist simply observes that the term ‘mental object’ is a meaningless term invented by Platonists to push the Platonist agenda; one can postulate that a certain arrangement of neurons is associated with the emergence of a mental concept, but that does not somehow conjure up any actual object - hence (iii) can also be summarily rejected.
In his section 6.4 Deflationary Truth and Modality, Balaguer makes the same mistake that I talk about on the page Natural language and Reality. He takes statements and considers them as though they could be independent of any language system, without reference to any language system that they might belong to, and the result unsurprisingly is nonsensical. At one point he says:
…consider these two sentences:
17 is the successor of 16.
16 doesn’t have a successor; rather, 16 is the largest number.
“Deflationary-truth nominalists claim that H1 is true and H2 is false. But, again, why? Given that they think there are no such things as numbers, and, in particular, no such things as 16 or 17. Why do they think H1 is true and H2 isn’t? ”
This is a classic straw-man construction. In fact, a non-Platonist has no reason to think that H1 is ‘true’ and H2 is ‘false’. But he can say that H1 is provable in Peano arithmetic and similar systems and the negation of H2 is also provable in Peano arithmetic and similar systems. (Footnote: One might note that for convenience, a non-Platonist might say off the cuff that H1 is ‘true’ as a shorthand to mean “provable in Peano arithmetic and similar systems”, and similarly for H2, without thereby committing to Platonism.) He could also say that a mathematical system where H1 applies and where H2 does not is a better system for the application to real world situations.
In section 8 Modal Nothingism to the Rescue, Balaguer spend several verbose pages trying to justify the notion that talking about whether things might be possible or not possible in real world scenarios has some relevance to mathematical systems. In doing so, he rehashes much of the material that I have already dealt with both in my Part 1 and on this page, repeating the same fundamental errors. But in any case the notion that it might be relevant is nonsense; in real world scenarios, the notion of ‘possible’ involves the notion that some aspects of the real world can change - that something could be possible if some aspects of the real world scenario were changed, and similarly, something might not be possible if certain aspects of the real world scenario were of a certain nature. But in well defined consistent mathematical systems, nothing can ever change, everything is fixed - if a statement is provable now, then it is provable in that system for all time. And that means that the assumption that an analysis of scenarios where there can be change is applicable to systems where there can be no change is completely wrong.
If the negation of some conjecture is provable in the system, then the conjecture is not ‘possible’, and if both the conjecture and its negation are not provable in the system, again the conjecture is not ‘possible’. On the other hand, perhaps the conjecture is provable in the system. But the key point here is that unless you know the provability or otherwise of the conjecture, you cannot speculate that it might be ‘possible’ - it is only ‘possible’ if it is provable in the system, so you could only postulate that it is ‘possible’ in the system if you already know that it is provable in the system - and if you know that it is provable in the system, the notion of it being ‘possible’ is utterly redundant.
Finally, here’s an ironically relevant comment from Mark Balaguer’s book Free Will: (Footnote: Mark Balaguer, Free will, MIT Press, 2014.)
“You should always beware of someone throwing a bunch of scientific studies at you and telling you that these studies establish X, Y, or Z. You can’t trust people on stuff like this. You have to read the journal articles yourself to see what they show. And if you don’t have time to do that, you should be skeptical. In short, you should remain unconvinced.”
Similarly, you should beware of someone telling you that - if you do not believe in the existence of things for which there is no evidential nor logical basis - then you must also back up that statement of non-belief with other notions, as prescribed by Mark Balaguer - regarding that lack of unfounded belief.
It should hardly need observing that that is complete nonsense. The onus rests, and will forever rest on those who claim that such things exist, to provide evidential or logical basis for those claims.
In the same way that I don’t have to provide any justification other than the absence of any evidential or logical basis for rejecting the notion of the existence of Santa Claus, or a Jewish god, or a Christian god, or unicorns, etc, etc… I don’t have to provide any justification other than the absence of any evidential or logical basis for rejecting the notion of the existence of independently existing real, non-physical, mathematical entities, and for rejecting the idea that that notion has any relevance to the advancement of mathematics.
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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