Could mathematical truths have been otherwise?

Question

Could at least some mathematical truths have been otherwise? Or are all mathematical truths necessarily true? For example, is it the case that "there exists only one complete ordered field up to isomorphism" could not have failed to hold? Personally, I believe all mathematical truths are necessarily true and could not have failed to be true, but I wonder, are there philosophers of mathematics who argue that certain mathematical truths could have failed to be true? And if there are such philosophers, can I read some of their writings?

Answer 1

Your question is way too imprecise. I personally believe that type∈type and obj∈obj and ∃x∈type ∀y∈obj ( y∈x ), where "type" refers the type of all types and "obj" refers to the type of all objects. However, the most obvious translation of this into the language of ZFC set theory yields something that can be disproven. You are not going to be able to perform very meaningful reasoning about the philosophy of mathematics until you are sufficiently precise, and that would tell you that you have used "truth" without defining it in any way. If you try to do so, then you will find that actually you are incapable of defining "truth" for statements beyond higher-order arithmetic, or maybe even for merely arithmetical sentences. You can define such "truth" within a foundational system (e.g. ZFC), of course, but that is meaningless here because you are talking about mathematics independent of foundations. (Saying that a theorem is necessarily true just because ZFC proves it is obviously 100% circular reasoning.)

Answer 2

I suspect this answer will give people headaches, but…

Mathematics is a reflection of our particular mode of perception and cognition: i.e., we perceive a world containing isolated objects, and thus invent numbers and basic numeric operations to count and track them; we perceive distance, area, and velocity in particular ways, and thus construct formulas to account for our perceptions.

So, any being that perceives the world the way we do will necessarily develop mathematical 'truths' similar to ours. But the way we perceive the world may not itself be necessary, and beings that perceive the world differently might develop quite different mathematical truths.

The universe itself has no mathematical truths, because the universe doesn't 'do' math; it merely is as it is. Mathematics and its truths are always developed by beings who analyze and interpret the world after the fact.

Answer 3

This question is profound and has been repeated so many times that I feel that there is desire to find relief in violation of reality which is assumed to be governed by mathematics.

Before I turn on to mathematics, let me explain the reality. There are various realms of existence. The realm in which we live is very robust mathematically. For example - A remains A but from Buddhist point of view A (no matter what it is ) is a phenomena. It is bound to change , arise and vanish. In religious text you will find more “magic” in which people believe that metal can be turned into gold or a whole universe can be created in six days, etc, or, in science ,the beginning is not understood and the beginning seems to be beyond mathematics or physics(which relies heavily on mathematics).

Coming back to Mathematics,Mathematics was a tool invented to solve certain problems in daily life(of this robust realm of existence). Then we went on to discover its broad range of applicability and in the process we discovered several in built properties and theorems of mathematics.

However once we cross this realm of robust existence the mathematics fails. It is consciousness driven. One can become many. Many can become one.

Even in this robust realm we find fuzzy mathematics which handles concepts like tall, handsome , speed etc in a fuzzy way. It is a valid mathematics but its operations are based on fuzzy concepts.

Asking could mathematical truths have been otherwise? is like asking are there different tools out there with its own set of truths ? The answer is Yes. One acceptable example is fuzzy mathematics. But the story does not end here , at the beginning of universe and in spirituality the reality itself is different and we have different rules to play with. (Note: Religious claims are unscientific )

Answer 4

There is not necessarily a strong consensus about this matter even among mathematicians who are also well-versed in philosophy (e.g. Hamkins, Koellner)^ℕ. Pluralism at various levels is balanced by a coherentism about how to co-justify the terms that are on those levels, so for example if the issue is type vs. set theory, the desire is less to have one framework "win out" over the other, and more to see how to "translate" set-talk back-and-forth into-and-from type-talk. This often comes down to showing that a given set theory has a moment of proof that is equivalent in use or value to a proof-moment in some type theory, rather than showing divergent results.

Still, pluralism would be much less "lively" if it were not at least sometimes an attempt to reconcile genuinely at-odds figures. So it is that Saharon Shelah, who is an extremely precise mainstream set theorist, has these "logical dreams" about extending forcing in some way even into "normal arithmetic" (if I'm reading him right, anyway). You could also take so-called "junk theorems" as samples of contingent mathematical truths, by saying that the "pointlessness" of 1 ∈ 3 vs. 1 ~∈ 3 is actually a glimpse into a pair of worlds, one of which is such that 1 is an element of 3 and the other such that 1 is not an element of 3, etc.

And I daresay that mathematicians are not likely altogether blind to the epistemological problems accompanying their work, so it may be fairly granted (unto them, that is) that they might not quite believe absolutely in the results of their proofs after all. A (classical) philosopher looks at a mathematical deduction as a holy miracle, if you will, a revelation of eternal verity; a mathematician always wants to make sure the proof is as crisp as possible, and is on the lookout for challenges and improvements by the by. The philosopher is impatient to "have" the eternal truth; the mathematician loves this truth too much to assume dominion over it for the sake of their own grandeur. (Or so a stereotype might go.)

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^ℕI should also emphasize the many recent, detailed contributions of Neil Barton, both in general and on the SE network itself. And so too Mikhail Katz contributes to some extent here, and a lot on the MathSE (among others).

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An overview/reframing: but the question admits of being refined, so that we might ask, "Are all mathematical questions such as admit of necessarily or contingently true answers, or does this vary?" So we might think that all the "qualitative" statements in mathematics are necessarily true, e.g. "Only one ordered field is complete up to isomorphism," or, "The zero ring has a Krull dimension of −∞," or, "The first measurable uncountable cardinal is also inaccessible." But then what about claiming that any of these things, these ordered fields or zero rings or uncountable cardinals, "actually" exist? Granting that they do exist, we might yet not think that there is an intrinsic logical necessity to this. So a whole class of mathematical claims (that rings, fields, sets, categories, types, etc. "exist" or not) might admit of contingency.

Answer 5

The mathematician Hans Freudenthal may have thought that mathematical truths were necessary - in some sense. He published an amusing book about how we could start communicating with aliens in a galaxy far away, Lincos (1960), starting with first counting in unary (and hoping they will start counting along).

Answer 6

As noted by Ted Wigley, mathematics appears to me to be constructed, not discovered "out there" like quarks, birds, viruses, etc.

That is not to say that there are more or less intuitively/empirically inspired mathematics, like basic arithmetic and geometry. A lot of very beautiful and elegant things were shown to hold based on axioms derived from our everyday experiences.

For example, the counting numbers are a useful way to generalize "amount" based on our ability to create boundaries between things. Note that these boundaries are somewhat arbitrary (albeit useful for survival) like "2 lions", "3 birds", "more tigers than humans" etc ;) But we shouldn't reify this ability to divide the world into the existence of some abstract "number stuff".

However, around 19th century parts of mathematics started getting creative with their axioms. You have non-Euclidean geometry, abstract algebra, quaternions, and things I cannot even describe they are so abstract and otherwordly (see: https://mathoverflow.net/questions/114787/what-is-teichm%C3%BCller-theory-and-its-history)

For every consistent set of axioms/properties you set out, you get some new mathematics. I emphasized the word consistent because we take classical logic as almost bedrock for abstractions in the world. Whatever else may come and go, good ol' logic must hold.

Unfortunately, even logic has succumbed to the creativity of the human mind: https://en.wikipedia.org/wiki/Outline_of_logic

As examples, we have quantum logic (although its not super useful in everyday world), paraconsistent logic, and, perhaps most well known, intuitionistic logic, each of which expands or modifies one or more elements of classical first order logic.

So, given a particular set of axioms and inference rules (i.e., logic), indeed the "truths" that pop out could not have been otherwise. However, I think people read way too much into the metaphysical import of the success of mathematics and its methods. It's really a highly abstract/rigorous extension of our ability to use language.

Answer 7

OP says:

Personally, I believe all mathematical truths are necessarily true and could not have failed to be true, but I wonder...

Seriously?!

One may discuss whether a truth is necessarily or contingently true under the stable assumption that it is true.

But if a putative truth is being debated between groups of mathematicians as true or not, your claim above is moot. You seem think math entirely consists "2+2=4" type stuff, i.e. it has no controversies that mathematicians hotly argue and disagree upon.

Below
[Section 1) Collection of some well known examples of hotly disputed claims within math to rebut this notion that math consists of stable eternal truths. Surely a practicing mathematician could give many more.
[Section 2.] The main question which amounts to being around foundations
[Section 3.] Some thoughts on platonism and Platonism wrt math.

1. Controversial Mathematics

Note: These are arguments within math, not with philosophers of math. So there's no way to call them True without taking a partisan side.

1. God made the natural numbers…: Kronecker vs Cantor
2. Infinity and The Axiom of Infinity:
   Gauss stated, I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking.
   Directly contradicted by Cantor's Axiom of Infinity: There exists an infinite set
   Who should we believe Cantor? Or Gauss?
   Its ironic that mathematicians — which includes the modern ones — use Gauss's researches in their everyday work but doctrinally believe Cantor.
   Is that correct?
3. Axiom of Choice: This controversial axiom in set theory states that for any set of non-empty sets, there exists a choice function that selects an element from each set. Various counterintuitive consequences follow (below). Yet mathematicians are loathe to give it up
4. Banach-Tarski Paradox: This theorem states that it is possible to decompose a solid ball into a finite number of non-overlapping pieces and then reassemble those pieces into two solid balls identical to the original. This paradoxical result relies heavily on the Axiom of Choice.
5. Non-Measurable Sets: The existence of sets that cannot be assigned a meaningful measure (volume) is a consequence of the Axiom of Choice, leading to ongoing debates about their legitimacy and implications.
6. Continuum Hypothesis: Proposed by Georg Cantor, this hypothesis states that there is no set whose size is strictly between that of the integers and the real numbers. Its independence from ZFC axioms means it can be neither proved nor disproved within that system, leading to significant controversy.
7. Ultrafinitism: A form of finitism that denies the existence of very large numbers and infinities even in principle, ultrafinitism challenges conventional views in mathematics and has led to intense debates.
8. Constructive Mathematics vs. Classical Mathematics: Many of the above can be summarized here:
   Constructive mathematics requires that existence proofs provide explicit examples. whereas classical mathematics allows non-constructive proofs. This leads to debates over the acceptability and usefulness of non-constructive methods.
9. The Four Color Theorem: Is a computer generated proof that runs to thousands of pages un-understood by humans a legitimate proof?
10. Zero-Knowledge Proofs: In cryptography and complexity theory, zero-knowledge proofs allow one party to prove to another that a statement is true without revealing any information beyond the validity of the statement itself. The mathematical foundations and implications of these proofs are hotly debated.
11. Independence of the Parallel Postulate: Currently not a hot topic. But imagine the discussion from Saccheri to Einstein — 300 years! Now come to the next:
12. Homotopy Type Theory: Can HoTT replace classic ZFC? Should it?

2. The question of Foundations

In the physical/empirical world, because there can only be one thing at one place-time, unique identity of objects is easy and taken for granted.

But for abstract objects it is not so easy. As Michael Carey asks:

What does it mean to you that a mathematical truth fails to be true? That it is independent of ZFC? That it is an inconstant definition? That we cannot use it to do anything??

We try to tighten and strengthen the discussion by establishing foundations.

But that gets into it's own difficulties as this old comment on math SE indicates:

Mathematics is really Psychology.
Psychology is really Biology.
Biology is really Chemistry.
Chemistry is really Physics.
Physics is really Mathematics.

Addendum:

And Computer Science is really recursive, so it's all these.

Notes:

1. "really" is anything but real(ly).
2. Expanding each really into something more substantive (and real!) would fill tomes and is left to the reader.
3. The first line of the first quote is really(!) pertinent to this question. But I felt a broad conspective view would be more helpful

For the current context, the main point about the math-SE comment is that explanations generally are invariably of some A in terms of a simpler or better known B, i.e. explanations are inherently reductive. But when one takes such reductions too far one can run in circles!

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Expansion added

In response to questions in comments suggesting that this comment is a joke therefore has no serious import.

Here is the expansion of each line.
With no pretensions... Done by chatGPT

1. Mathematics is really Psychology.
   Intuitionism: L.E.J. Brouwer's philosophy of mathematical truths as essentially the stuff mental constructions — intuitions (Kant's word). Also Poincare's rejection of logicism
2. Psychology is really Biology.
   Cognitive Neuroscience: Eric Kandel's work on the biological basis of learning and memory.
3. Biology is really Chemistry.
   Molecular Biology: James Watson and Francis Crick's discovery of DNA structure.
4. Chemistry is really Physics.
   Quantum Chemistry: Linus Pauling's application of quantum mechanics to chemical bonding.
5. Physics is really Mathematics.
   Theoretical Physics: Albert Einstein's use of mathematics to develop the theory of relativity.
6. And Computer Science is really recursive, so it's all these.
   Recursion and Alan Turing: Alan Turing's work on the Turing machine, which is foundational to computer science and illustrates the recursive nature of algorithms and computation.

[If you ask me I dont think chatGPT got 6 at all]

3. Platonism and platonism in Math

Added later in response to comments like "Just some old mans's":

It seems most answers given thus far conveniently assume mathematical realism and Platonism are incorrect without providing any deep insight as to why. The jury is certainly not out on the issue. I suggest you refer to the SEP for deeper insight.

[AFAIC]:

Every mathematician is a platonist.

This is universally true ranging from the schoolchild who understands that there are right and wrong answers independent of what the teacher says to the greatest of mathematicians who can correct a result they earlier got wrong.

All that is needed to be a platonist is to acknowledge that true/false are not just notions in the boolean object world he may play around with but are invariably present in the working meta-langauge that the mathematician habitually swims in.

But then the objection will be raised: Don't a number of mathematicians dispute platonism? (And and even larger number of them have no interest in the question)?

To answer this I use a nuance that comes from Conifold:

• Platonism (capital P) is the philosophy of Plato (a guy!) Who happens to have spoken on a variety of subjects, including the philosophy of mathematics. That person happens to have died a while ago so we use "Platonist" to assert that the philosopher/mathematician in question tries to align with Plato — to the best of 2024 lights given that we cannot consult him directly
• platonism (small p) is a modern philosophy of mathematics associated with figures like Frege, Cantor, Gödel and so on and centers round a commitment to "abstract objects".

I (personally) seriously doubt whether Plato — the philosopher — would have

• approved of calling entities in his "world of form" abstract objects
• sided with Cantor over Gauss in carelessly ascribing reality to infinity
• In his time, Cantor's work was suspect as being more theology than math.

IOW platonism is at best orthogonal to Platonism. IMHO it's > 90°.

More in other answers such as What is it that is done when we DO mathematics?

Answer 8

In all conceivable universes governed by physical laws mathematical laws also work.

But one can conceive a universe (flow of qualia) where mathematical laws do not work. For example, a universe that you occur in a night dreem. In that universe mathematics and even logic can be different.

One can conceive (for instance) a universe where all what the obserer believes is true, or where all the observer wishes is true, or where all the observer fears is true. In such universes the logical paradoxes are inevitable, nevertheless such universes are conceivable. As viewed from the inside, the mathematical laws could be broken there.

So, the mathematical laws are not empirical in the sense that one needs scientific method to verify them (because scientific methods requires mathematics already to work in the first place), but empirical from the point of view of a given observer.