Mathematical truth What makes mathematical theorems true
Can mathematical theorems be true? Balaguer's position on the existence of mathematical objects
Table of Contents
Introduction: what makes a sentence true?
Balaguers project and the goal of housework
FBP and fictionalism
The epistemic problem
Defense of FBP
What is left?
Introduction: what makes a sentence true?
3+2=2+3 or Pi is an irrational number seem to be true sentences at first glance. Our pre-theoretical assumption is mostly that the usual mathematical propositions, especially definitional propositions, are trivial truths. We trust mathematics so much that we constantly incorporate it into the theoretical rationale behind our best scientific theories. But many a philosopher of language would intervene here and claim that mathematical propositions are anything but trivial truths. To say that 3+ 2=2+3 is a true statement, can be very problematic from a language-philosophical point of view and can quickly lead to strange results. Because if one asserts the truth of such a sentence, then one is committed to a lot more at the same time. Let us first ask ourselves what makes an everyday verbal and non-mathematical sentence true. So we are now looking for the so-called truth makers of sentences. The author of this paper is a student at the University of Duisburg-Essen. This sentence appears to be true for two reasons. On the one hand because there is an author of this term paper (namely I, Petar Santini) and on the other hand because the author of this term paper is actually a student at the University of Duisburg-Essen. Or alternatively expressed in the vocabulary of the famous language philosopher Frege: The singular term the author of this term paper has a meaning and this singular term comes with the predicate is a student at the University of Duisburg-Essen to. The word meaning in relation to singular terms has to be understood differently than in the everyday sense. If a singular term has a meaning in the Fregeschen sense, then it means that a reference object exists to which this singular term refers. The truth of a proposition does not seem to depend only on the fact that a certain object is assigned a correct property.1 The object also has to exist. Another candidate that can make sentences true is existence. So let's go back to our original example 2+3= 3+2. Why is this sentence true? Is the correct predicate assigned to the individual terms or is a correct relation established between the numbers? Trivially, this question can be answered with the following argument with a Yes answer:
P1: The set of real numbers form a group with the arithmetic operation +
P2: In a group with arithmetic operation +, commutativity applies to all of its elements:
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P3: 2 and 3 are elements of group (R, +)
Figure not included in this excerpt
The algebraic axioms solve this problem, but what about the condition of existence? Do numbers exist? Do math objects exist? This is where the so-called Platonists come into play. They claim that mathematical objects exist. In fact, they exist as abstract objects. The question now arises, what are abstract objects? In short, the Platonist would characterize them as non-space-time objects. They do not arise, they do not disappear and they are not causally effective. A Platonist is thus faced with the challenge of establishing an ontological theory for abstract objects. He must be able to explain why abstract objects exist and he must be able to explain their way of being. The short and concise answer of a Platonist to the initial question is: The truth of the statement 2+3=3+2 follows from the algebraic axioms and from the existence of real numbers.
But as is now typical of philosophy, everything is a matter of dispute. And it is precisely in this context that the opposing party is introduced: the so-called anti-Platonists. They refrain from postulating entities like abstract objects and pursue a different strategy. A variety of anti-platonists are also called fictionalists for this reason, as they regard mathematical objects as fictional objects. Mathematical objects, according to the fictionalists, are fiction because they do not exist. For this reason, they do not need to propose an ontological theory about abstract objects. Instead, they resort to paraphrase strategies and a hermeneutics designed for mathematics. The problem that arises with mathematical sentences should, so to speak, be analyzed away by taking an appropriate standard reading for mathematical sentences as a basis. Accordingly, the sentence 2+3 = 3+2 be read differently than before and be paraphrased. But what good is a paraphrase if the truth of a sentence depends on the existence of the objects to which the linguistic expressions that appear in it refer. It is completely irrelevant how the sentence is paraphrased. As long as numbers appear in a sentence and at the same time, like a fictionalist, one asserts that numbers do not exist, then the sentence must be wrong. Here again many a philosopher of language would speak up and defend the anti-Platonist. In fact, there are sentences in which the truth of the propositional sentence in no way depends on whether the terms in it have a meaning (i.e. have an existing reference object to which they refer).2 The sentence Harry Potter wears glasses carries the truth value incorrectly since Harry Potter does not exist. The sentence Jonas believes that Harry Potter wears glasses, would again be a true statement since it is a report of mental states. Operators like X thinks, X claims etc. ensure that the sentence is correct exactly when X is assigned the correct belief or the correct assertion. The truth of the attribution of belief does not therefore depend on whether Harry Potter exists, but depends on Jonas actually believing that Harry Potter wears glasses. For this reason one speaks of existence-indifferent operators. And it is precisely with such existence-indifferent operators that the fictionalists also work. Accordingly, the sentence says 2+3 = 3+2 nothing else than: Within the history / theory of mathematics, the following applies 2+3 = 3+2 is. So here a fiction operator is used, as it could also be used with sentences when talking about a series or a fantasy novel. What does that mean now? Then mathematical propositions are not true in both cases. Both the platonists and the fictionalists? Of course, that's not entirely true. When a person makes the assertion that 2+3= 3+2 If you really mean it, then the Platonist would nod and agree, but the Anti-Platonist would say: "Yes, but that's only true within the history of mathematics." Mathematical theories would therefore be treated in the same way as a fantasy novel. So there is an essential difference between the two parties: Mathematical sentences taken by themselves and without an existence-indifferent operator are always false statements for fictionalists.
Balaguers project and the goal of housework
Which of the two theories is the right one? Are we even able to find out who is right? Is there an argument that we have not yet figured out which one of the two theories refutes? Mark Balaguer dealt with these questions in his work Platonism and Anti-Platonism in mathematics3 and tried to answer them. Balaguer's work and his argumentation in relation to the questions mentioned are the main subject of this paper. Already at the beginning of his work, Balaguer makes it clear which project and which goals he is pursuing. He would like to prove the following 3 conclusions in his work:
a) It should be shown that both Platonism and Anti-Platonism can be defended against all previous objections and that we therefore currently have no way of falsifying (or verifying) one of the two theories. He calls this conclusion "the weak epistemic conclusion"4 (weak epistemic conclusion)
b) It should be shown that we are not only current have no way of falsifying (or verifying) one of the two theories, but that in general it is impossible for us to construct an argument from which it follows that mathematical objects exist or do not exist. Balaguer calls this conclusion "Strong epistemic conclusions"5 (strong epistemic conclusion)
c) It should be shown that not only because of our mental and physiological limitations we have no way of finding out whether one of the two parties is right, but that there are no facts at all in our world to check whether abstract objects exist or Not. Balaguer titled this conclusion with the designation "Strong metaphysical conclusion"6
A full analysis of all 3 arguments would go beyond the scope of this paper, therefore the main focus in the following will be on a) and b), whereby in a) only Balaguer's defense of Platonism against traditional objections will be dealt with:
FBP and fictionalism
According to a) Balaguer wants to prove that both Platonism and Anti-Platonism could survive all traditional objections and therefore there are currently no arguments to refute either of the two theories. For reasons of clarity, the goal of the argument was not described very precisely in the previous chapter. The fact is that there is not only one form of Platonism, and it is not the case that there is only one form of Anti-Platonism. In fact, there are many variants and degenerations of the two theories, which Balaguer also considers false and therefore does not defend. His project is about defending certain forms of these two currents of thought. With regard to Platonism, he would like the "full-blooded platonism"7 (FBP), translated to defend thoroughbred Platonism. When Balaguer speaks of defend, he means two things. On the one hand he wants to show that all other forms of Platonism that are not FBP are wrong and on the other hand he wants to show that FBP outlives all traditional objections. Regarding anti-Platonism, he wants to defend fictionalism. Its defense is analogous to that of Platonism, i.e. all variants of anti-Platonism are falsified and fictionalism is defended against the traditional objections of the Platonists.8
Now what is FBP? And what is fictionalism? Let's start with full-blooded Platonism, or FBP for short. Nowadays there are all the mathematical constructs and sets that were once introduced. The traditional Platonist would argue that all of these sets and structures exist as abstract objects. But there was also a time when complex numbers, for example, were not yet introduced. There was a time before Euclidean geometry was invented and there was also a time when circles or squares were not yet defined. There will probably be more mathematicians in the coming centuries who will define more sets. Do these sets already exist? Are there also possible mathematical sets that are never defined by a human being? Another question that Platonists have to ask is whether abstract objects also exist independently of us and our mathematical theories. And this is where Balaguer's distinction comes into play. He differentiates between perfect, thoroughbred Platonism and imperfect Platonism. As the names suggest, the full-blooded Platonism (FBP) is the theory which postulates all logically possible mathematical objects, while the imperfect Platonists only claim the existence of certain mathematical objects and in turn describe other mathematical objects as metphysically impossible.9 So if we think back to the introductory questions of this chapter, there could well be some sort of imperfect Platonists who tie the existence of mathematical objects to active thought and construction processes of a mathematician. In the course of this distinction, Balaguer tries to logically formalize FBP and runs into some problems. In a first attempt, Balaguer tries to formalize FBP in this way: Vx (x is a mathematical object & x is logically possible) x exists)10
The two core problems of such a formalization of FBP is, one, that existence is used as a predicate. If one takes a closer look at various ontological proofs of God, this can lead to absurd conclusions. A philosopher who recognized this problem was Kant. If one sets premises within an argument, these premises can be true for different reasons. These are either analytical judgments or premises, which we also frequently call analytical truths, or synthetic judgments or premises of those typically before Kant Critique of Pure Reason it was thought that they were always based on empirical knowledge.11 Analytical judgments a priori can be refuted by showing that the term was misunderstood or incorrectly deduced. For example, synthetic judgments a posteriori can be refuted or confirmed by empirical evidence. The sentence a bachelor is unmarried is analytically true. This can also be seen from the fact that the sentence There are bachelors who are married creates a contradiction. The predicate for a concrete object x exists however, ascribing can never be true for analytical reasons. According to Kant, existential ascriptions are always synthetic a posteriori judgments that are always based in some form on empirical judgments.12x is logically possible and x is a mathematical object, but x does not exist, does not seem to represent a logical contradiction and for this reason it cannot be the case that existence is derived from the concept logically possible mathematical object is deducible. This problem can hardly be dealt with through alternative formalizations. Whether FBP can defend itself against this objection will become clear in the course of the housework. Another problem is the use of the term logically possible. According to Balaguer, there is one party that claims that what is logically possible actually already exists in our universe. While the other party claims that no factual existence of the possible entity follows from the logical possibility of an entity.13 Balaguer's intuitive understanding of logical possibility could be reconstructed as follows: An event that is logically possible is characterized by the fact that it has a probability that is greater than zero. Every event that is logically impossible and therefore contradictory has a probability of zero. So the event of a one-time coin toss with a single coin is logically impossible to roll heads and tails at the same time. The logical possibility can thus be characterized by consistency, but it does not yet imply, at least in Balaguer's understanding, an existence.14 The only objects for which logical possibility also guarantees existence are mathermatic objects. At this point, like Balaguer, I would like to be satisfied with the fact that FBP cannot be formalized completely, or rather, not free from problems, and be satisfied with an intuitive and informal understanding of this theory: all mathematical objects, systems or sets die one could ever think up and are coherent, i.e. free of contradictions, are logically possible and thus exist. From the point of view of FBPs, a number that is a natural number and is negative at the same time cannot exist, since this is a number that is logically impossible, since natural numbers cannot be negative by definition. This is my intuitive understanding of Balaguer's FBP and it is precisely this understanding that is used in the course of my argument reconstruction and analysis.
Now we come to the already mentioned fictionalism. As already stated in the introduction, this states that mathematical theorems like 2+3=3+2 are only true if one works with existence-indifferent operators.The sentence 2 + 3 = 3 + 2 is wrong, but the sentence In the history / theory of mathematics, 2 + 3 = 3 + 2 is true. Or, in other words, fictionalists do not believe in the existence of abstract objects and therefore work with different truth conditions for mathematical theorems.15 In addition to fictionalism, there are many other variants of anti-Platonism, which Balaguer carefully names in the introductory chapters, but these are no longer relevant or worth mentioning for the present discussion.16
1 Thank God, Frege. (1892). About meaning and meaning. In: Voigt, Uwe (ed.): Reclam Great Papers Philosophy. Ditzingen: Reclam, 2019
2 See Frege 1892.
3 Balaguer, Mark. (2001). Platonism and anti-Platonism in mathematics. Oxford [et al.]: Oxford University Press.
4 Balaguer 2001, p.17.
7 Ibid., 5.
8 See Balaguer 2001, pp. 14-15.
9 See ibid., P. 5.
10 See ibid., 6.
11 Kant introduced the category of synthetic judgments a priori.
12 See Kant, I., & Erdmann, B. (2011). Critique of pure reason: [main volume] (5th continuously rev. Ed. Reprint 2010 ed.). Berlin: Georg Andreas Reimer Verlag, De Gruyter, pp 461-462.
13 See Balaguer, p.6.
14 Except for mathematical subjects
15 See Balaguer, Mark. (2001). Platonism and anti-Platonism in mathematics. Oxford [et al.]: Oxford University Press. P.13.
16 See Balaguer, Mark. (2001). Platonism and anti-Platonism in mathematics. Oxford [et al.]: Oxford University Press. P.11.
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