# How could mathematics and logic exist without us, if they are concepts created by us independent of reality?

Would maths and logic exist if we didn't exist despite we created them, and do not have correspondence with reality?

• There are two different things that "maths and logic" can mean just as "physics" can refer to physical objects or our theories about them. The latter can not exist without us, the former can (on the common sense view). There is no conclusive argument as to whether mathematical and logical entities similarly exist, but there is none against it either. So they might, and what we created is just an imperfect reflection of them. Jan 18, 2018 at 21:35
• I think this question should be rephrased, as it presupposes we created math and logic. Jan 24, 2018 at 18:23

Take inverse square laws. You can see them as mathematical or relying on logic, but they are geometrical and relational and just a part of the beingness of things. The ratios involved come from the conditions for there to be anything rather than nothing, we think.

Vs infinity. A really useful mental tool, but which never exists in the world. Exactly defining pi, and being able to develop differentiation, rely on imagining series' to be infinite. They help us mentally transfer between contexts, linear & circular, discrete & continuous.

So we have things that depend on counting, which depend on the properties of solid bodies at roughly our scale & in roughly our environment (not eg. quantum scale or the surface of a neutron star), which do absolutely exist, in specific instances, without us in the world. Then we have generalisations and abstractions of these, which don't; along with idealisations like infinity and perfectly round circles, which also never exist in the world.

Logic helps us organise our experiences, and it exists like a mental constellation around the star of our own concerns. But there are other stars, a whole universe of galaxies of other ways to think and organise experiences.

• +1:It's often forgotten, given how commonly Einsteins General Relativity is seen as the geometrical theory par excellence, that there are geometrical elements in Newtonian gravity & Maxwells EM - as you've said, 'the inverse square laws ... are geometrical', the square arising from the dimension of a sphere surrounding a gravitating or electrical source. Jan 18, 2018 at 16:23
• @draks: Good question. I forgot who said that it was a marvel that we find the physical universe can be so described. It's forgotten that when physics got started it was done conceptually and not mathematically, though there were elements of mathematics in it, as in the Timeaus. I don't think that the universe can be wholly described mathematically, as we are conscious, and we are a part of the universe and mathematics has no purchase on consciousness despite models such as Markov brains and neural networks. Jan 19, 2018 at 12:40
• Perhaps, mathematics is the aspect of neccesity in the world, and consciousness the aspect of freedom. Jan 19, 2018 at 12:41
• @draks... I believe reality is consistent (or at least, I accept that as a working hypothesis) but not redundant, meaning it can be modeled but not completely reduced to a subset of itself. In other words, I believe a complete and consistent theory of everything is impossible. Jan 20, 2018 at 7:03
• Jan 20, 2018 at 11:41

There are two basic positions:

Platonism which asserts a world of ideal forms and numbers, geometry and mathematics, ideally are part of that world. Mathematics is then a discovery and not an invention. Aristotle for example, assents to this immediately in his Categories. Although it was a widely held position in the pre-modern era, It's much less common today.

Constructivism is the position that you yourself are indicating. Here, mathematics is conceived by men and so is invented and not a discovery. This is the usual position in the modern era. Though, one category theorist, Eugenia Cheng, explained that when she spoke to philosophers she was persuaded that they were constructed but when she returned to thinking about mathematics she was convinced that they had a real existence.

Hegel famously invented a form of logic that was dynamic and organic. It arises in the world and molds the world. He called it the dialectic as an allusion to the Eleatic Monists from Ionia that first began to elaborate such a logic. (It also has a clear resemblance to a dialectic elaborated in the Tao). This is far from our modern notion of logic which is purely syntactical and formal.

One way of thinking about this is to see that logic is a form of necessity. And for Hegel, his Logic of Nature has that aspect, it too is a form of necessity. The closest analogue to this today are the natural laws of physics, which are the necessary laws of nature herself.

Physicists sometimes speak of discovering the laws of nature in the pure form of necessity. In this pure form, there would be nothing contingent in it. For example, in the Standard Model of Particle Physics there are some thirty free parameters. This is contingent. To reduce this number is the aim of some physicists.

• I’d say that Platonism (in some form) is still the most widely held view. I’m counting here set-theoretic reductionism and varieties of structuralism like Stewart Shapiro’s ante rem structuralism which postulate abstract mathematical objects, albeit differing from naive Platonism. It’s certainly true, however, that Platonism is much more varied than the traditional position and that there are competitors. (But note, too, that the Platonism vs. Nominalism debate existed in medieval times as well; Ockham being the most famous proponent of nominalism.) Jan 18, 2018 at 18:07
• @Dennis: Perhaps, I haven't looked seriously into it. But that seems to me that a Platonist view is at odds with a secular, materialist view which is very widespread; and this is what I was basing my judgement on. Jan 18, 2018 at 18:26
• You're confusing constructivism with intuitionism: constructivism says "proofs have to be constructive", intuitionism says that mathematics is a product of the human mind. You're also incorrect in asserting that intuitionism, which is what you actually mean, is the dominate view. This seems to be related to the point I tried to share with you on your other question where it seemed that you believe that platonism is the only form of realism and that you confused intuitionism with nominalism. It's also misleading and incorrect to say that platonism and constructivism are the two 'basic' views. Jan 19, 2018 at 7:22
• @MoziburUllah: I must agree with Not_here that it's wrong to describe constructivism in mathematics the way you did: "Constructivism is the position ... mathematics is conceived by men and so is invented and not a discovery." Amongst mathematicians, at least, 'constructivism' is not about the 'discovery' vs. 'invention' debate, but (as Not_here noted) about types of acceptable proof. Specifically, in mathematical constructivism, proofs of the form "assume not P, contradiction, therefore P" without actually showing how to construct a P are not acceptable. [more] Jan 20, 2018 at 9:05
• [continued] It sounds like you're maybe confusing mathematical constructivism with constructivist epistemology. - As a side note, Intuitionistic logic is a form of constructive logic, but not all constructive logics are intuitionistic; nonetheless, many people confusingly conflate the two. Jan 20, 2018 at 9:08

There are three possible positions on this subject, held respectively by strict Platonists (are there actually any today?), theists, and atheists [1]:

• Strict Platonism: self-consistent formal systems exist really in a world of pure forms from all eternity, and temporally in the minds of people who discover them.

• Theistic Platonism: self-consistent formal systems exist virtually in God from all eternity, and temporally in the minds of people who discover them.

• Fictionalism: self-consistent formal systems exist virtually only in the minds of the people who build them or learn of them, just as the plot of a novel exists virtually only in the minds of its author and its readers.

"Discover" in both flavors of Platonism and "build" in fictionalism is just the same activity viewed in different perspectives. Everyone agrees that mathematicians can develop only those self-consistent formal systems that can exist, and in that sense they may be said to "discover" them. However, for an atheist those formal systems existed nowhere in any way before being "discovered", and therefore they are strictly "built" by mathematicians.

Further, both Platonism and fictionalism can be "plenitudinous" [1], meaning that all self-consistent formal systems are on an equal standing, so that

• Euclidean geometry is no less or more "real", or "true", than elliptic or hyperbolic geometry as formal systems, and

• (ZFC + CH) is no less or more "real", or "true", than (ZFC + ¬CH) as formal systems [2].

References/Notes

[1] Balaguer, M., 1998. "Platonism and Anti-Platonism in Mathematics". https://books.google.com/books?id=UEyPF1T6EbUC

To note, Balaguer's plenitudinous Platonism is equivalent to Resnik's and Shapiro's ante rem structuralism:

Resnik, M., 1997. "Mathematics as a Science of Patterns". https://books.google.com/books?id=EU2G_BFt7YsC

Shapiro, S., 1997. "Philosophy of Mathematics: Structure and Ontology". https://books.google.com/books?id=9xVErjy9qPQC

[2] ZFC = Zermelo–Fraenkel set theory with the axiom of choice. CH = continuum hypothesis.

Kurt Gödel showed in 1940 that CH cannot be disproved from ZFC.

Paul Cohen showed in 1963 that CH cannot be proved from ZFC.

Therefore, if ZFC is consistent, then (ZFC + CH) and (ZFC + ¬CH) are also consistent.

Created or invented, it's hard to deny that "2 + 2" will always equal "4", as it is a fact of logical necessity; or that Pythagoras's theorem or the prime number theorem will always be true no matter if there wasn't another human being around to prove it to themself. In that sense, mathematical truths do not depend one iota on empirical reality, but whether these truths would 'mean' anything without humans around, is an entirely meaningless question. We need to accept the axioms, rules, and syntax that a mathematical statement is based upon to state anything at all. However, correspondence with reality has no bearing, whatsoever, on what mathematical knowledge uncovers. It's similar to how the rules of the game of chess do not depend on empirical reality to be played. In an alternative universe, you could imagine the game of chess being played with the same rules that we obey in our universe. Analogously, that's why it's interesting to consider how mathematics can be not only universal but contain truths relating to more than just our universe and all possible worlds.

That said, there are some instances of mathematics being advanced or challenged by what is understood about empirical reality, such as quantum logic. Our knowledge of pure mathematics may be advanced first, then an application opens up in the universe around us. Non-euclidian geometry and speculations about higher dimensional geometry in string theory are examples. Or, on the other hand, and upon reflection of history, when the calculus was invented by Leibniz and Newton, independently, it was for its direct application in physics and so on. The "unreasonable effectiveness" of explaining empirical reality beyond the original applications of these mathematical tools almost always greatly and surprisingly surpasses their original applications.

• "it's hard to deny that '2 + 2' will always equal '4' - Well, not necessarily. In modular arithmetic, and in particular, in Z/2Z, it can equal '0'. And not only is the continuum hypothesis independent of the axioms of ZFC, we can actually mathematically prove it to be both 'true' or 'false' by using the technique of 'forcing'; the paper "The Set-Theoretic Multiverse" has details. [more] Jan 20, 2018 at 9:25
• [continued] Still, all this just further supports your comment that "[w]e need to accept the axioms, rules, and syntax that a mathematical statement is based upon to state anything at all." Jan 20, 2018 at 9:25
• The language may be different, but take a thing, add another thing, how many things do you have? The answer will always be equivalent to what we understand as '2 things'. Another example would be that is impossible to arrange those 2 things in a square of equal length sides, regardless of whether you know about the mathematics of prime numbers. Jan 22, 2018 at 0:46

The apparent inconsistency that you are expounding, is the result of not realizing that there are two types of "mathematics and logic" - and treating them as one!

There is the "natural" math/logic, and the "man-made" math/logic.
The first type "exists," independent of "us," because the Universe exists.
The second type "exists" because it was created by "us."

Therefore, if your question refers to the first type, they do "exist without us." If it refers to the second type, they do not!