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I’m inclined to think that if someone says mathematical truth is empirical, it would trivialize the meaning of “empirical”.

For example, to say that we do not know a priori the answer to certain math questions, and have to discover them through a kind of trial and error, with recourse to some external concept of Truth in the universe, would mean that there is basically nothing that is not “empirical”, I think.

I think the important thing is that you can never know the answer to the question “Is there a coffee kettle on my kitchen table?” without observing a contingent state of the world; whereas you can solve mathematical questions purely by thinking (hopefully). But then, is that distinction superficial? One requires accessing physically non-present information (looking inside my kitchen); the other requires accessing mentally non-present information (figuring out how premises and rules lead to some consequences).

Then there is no difference in form, between the two (as in, that we must access inaccessible knowledge); but only in modality (one is “physical”, one is “logical”, but that distinction is insignificant).


It seems that the distinction between “empirical” and “logical” could be re-expressed as “contingent” vs. “necessary”; or “present as an immediate object of perception” vs. “a matter of reasoning and/or belief”.

Is there any good reason to say math is empirical, and if so, what does that mean “empirical” means, and what does that imply “not empirical” would be?

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Mathematics is not empirical: A mathematical proof is independent from any experience. It only relies on the correct application of the usual or a refined logic. To check new and original ideas on a field which is well-specified by precise definitions and axioms.

If someone considers mathematics to be an empirical science, it would be interesting to hear the arguments for this objection.

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    Math is independent of experience Seriously?! I for example have memories of math (and physics) teachers scolding me for solving problems in «disapproved» ways. Was that not an experience? Here is a more non trivial eg of a young Russian learning that the infinite set of all triangles pre exist — in short platonism — through a curt Shut up!
    – Rushi
    Apr 21 at 5:40
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    @Rushi The mathematical proof is independent from experience. I do not consider the experience when presenting your proof to other people. - Do you understand chapter 2 of Gurevich's paper from your link?
    – Jo Wehler
    Apr 21 at 8:17
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    We clearly have models of the mathematics in our head and manipulate them to arrive at new conclusions. I guess one way to look at the question is to consider the "physicality" or "reality" of these mental models. They are certainly more "malleable" than e.g. geometric paper cutouts; we can easily make mistakes. But those could be ascribed to faulty models or faulty operation. Essentially, the question is whether our mind is able to be sufficiently separate into different "agents" which can watch each other, so that one of them can be said to have an "experience". Apr 21 at 8:28
  • I would point out that this "concept of self-observing mental simulations" is actually one commonly accepted (I think) key feature of intelligence: Planning. We play (usually more mundane) scenarios in our minds (i.e., simulate them) and "observe" their outcomes. We use that to predict the future and direct our actions. We make plans. That can be social interactions, contingency plans for severe weather, a shelf we plan to install etc., or, relevant here, some math. In math we make models of things that don't exist in the material world. Is that a key difference? Apr 21 at 8:37
  • I'm a fan of this answer. Mathematics is a tool of science - it is not science, which requires the ability to experiment with repeatable results to demonstrate a theory is sound. Mathematics can reflect or express the truth of an observation (albeit imperfectly in many cases) but cannot prove empirically that something unobserved (e.g., dark matter, square wormholes, etc.) exists. Frankly, what truth is there to mathematics when the favored summer science fair class in my youth was "Lying with Statistics?"
    – JBH
    Apr 22 at 17:24
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We can say that there are many questions whose answers coincide with the answers of mathematics, that are empirical.

By an "empirical question," I mean a question that can be answered by some physical experiment or observation, or a question that is about a property of some physical object.

A mathematician's brain is a physical system, like a pendulum or mudslide, whose behavior we can investigate empirically. "Will the mathematician's brain conclude theorem X is true?" is therefore an empirical question about a physical system. It can be answered by an experiment of asking the mathematician to think about theorem X for some period of time.

The same is true of a computer. Computers are a bit more reliable than mathematicians when it comes to checking formal proofs, and they are of course physical systems that we can investigate empirically. "Will the computer agree that the mathematician's proof of theorem X is correct?" is thus an empirical question about a physical system. We can run an experiment, where we set up initial conditions of the computer program, allow it to run, and observe the result. It's a physical experiment about a physical object (the computer).

Then there are questions like, "Is there a solution to this Sudoku?" Now, this Sudoku is a physical object, ink on paper. To ask if there is a solution to it is to ask if there is a way to add ink to the paper, following the rules of Sudoku, until every square has a number. If the answer is yes, the yes answer is a property of a physical object, namely, the ink on paper (in the initial puzzle with some squares blank). Certain physical objects of ink-on-paper are Sudoku puzzles with solutions, and certain other physical objects of ink-on-paper are Sudoku puzzles without solutions. So "Is there a solution to this Sudoku?" is also an empirical question, because it is about a property of a physical object.

Then we may ask questions like, "In ZFC, is there a proof of the Riemann hypothesis?" This is not too different from the Sudoku. In Sudoku we have rules for when we can write a number in a square, based on the other numbers around it. In ZFC we have rules for when we can write a certain line of glyphs on a sheet of paper, based on the glyphs already written above that line on that sheet of paper. So, "In ZFC, is there a proof of the Riemann hypothesis?" corresponds to an empirical question about a property of a sheet of paper, just as with the Sudoku question.

So, whether or not you wish to say that mathematics itself is empirical, you should admit that essentially every mathematical question corresponds to an empirical question, such that the answer to the mathematical question is "yes" if and only if the answer to the empirical question is "yes."


Addressing some comments, I see that some people doubt that solvability of a Sudoku puzzle is a physical property of the sheet of paper + ink.

First, can we say it is a physical property of a sheet of paper, that the sheet of paper is black? (yes)

Can we say it is a physical property of a sheet of paper, that the sheet of paper is white? (yes, for a given objective specification of which reflected wavelength combinations are "white")

Can we say it is a physical property of a sheet of paper, that the sheet of paper is 99% white with 1% of its surface covered in black ink? (yes; two colors presents no more problem than one)

Can we say that it is a physical property of a sheet of paper, that the sheet of paper is 99% white, 1% black, and the black ink is arranged in a dot in the center? (yes; no problem to say the ink is arranged in a particular shape)

Can we say that it is a physical property of a sheet of paper, that the sheet of paper is 99% white, 1% black, and the black ink is arranged in a 9x9 grid? (yes; no problem to make the shape more complicated)

Can we say that it is a physical property of a sheet of paper, that the sheet of paper is 99% white, 1% black, and the black ink is arranged in a 9x9 grid with some of the ink arranged in the shape of an Arabic numeral 4 in one of the grid cells? (yes; no problem to have the shape of a numeral as well as the shape of a grid)

Can we say that it is a physical property of a sheet of paper, that the sheet of paper is 99% white, 1% black, and the black ink is arranged in a 9x9 grid with either the Arabic numeral 4 in one of the grid cells, or the Arabic numeral 8 in one of the grid cells? (yes)

So now we say that it is a physical property to be in one of a set of possible states of the paper: a 4 in the grid, or an 8 in the grid.

Now consider the set of all possible states of the paper that correspond to solvable Sudoku puzzles. Call it S. Can we say it is a physical property of a sheet of paper that it has ink arranged in one of the patterns of S?

Yes!

So, it is a physical property of a sheet of paper, that the paper is a solvable Sudoku puzzle.

A physical property of an object is a fact about what would happen (subjunctively) if a certain test, following certain rules, were applied to the object. The object has the property regardless of whether the test is in fact applied, and regardless of whether anyone capable of performing the test happens to be around.

Here are some examples of physical properties. The point of this list is so that, if you doubt that solvability of a Sudoku puzzle is a physical property of the Sudoku puzzle, you will need to find a point in the list where you draw the line and why; and the list is arranged to make it difficult to draw any such line.

  • The fact that a circuit has 5 Amperes of current flowing through it. There are many ways to test this. One possible test would be to correctly apply an ammeter to the circuit and read the display, following the correct rules of how to apply the ammeter.
  • The fact that a lawnmower is able to be started by pulling the cord. (without taking it apart and fixing it, without using some special device, etc.) Here, the test would be to pull the cord.
  • The fact that a Rubik's cube, with certain coloration of the faces, is able to be solved in the usual manner (without painting the faces, disassembling it, etc.) Here, the test would be to demonstrate such a solution that follows the rules.
  • The fact that a Sudoku puzzle can be solved in the usual manner (without erasing or crossing out numbers, etc.) Here, the test would be to demonstrate such a solution that follows the rules.
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    Your statement '"In ZFC, is there a proof of the Riemann hypothesis?" corresponds to an empirical question about a property of a sheet of paper,...' seems to miss the difference between content and the carrier of content.
    – Jo Wehler
    Apr 21 at 6:13
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    @JoWehler The point is that while you can perhaps interpret the statement in an abstract way that the paper is only a carrier for, you can also interpret it directly as a physical property of the paper. Just as certain sheets of paper are solvable Sudoku puzzles and other sheets of paper are not, and certain Rubik's cubes are solvable and others are not, depending on the coloration of the faces. Would you agree that the solvability of a Rubik's cube is a physical property of the cube? In the same way that whether a lawnmower can be started is a physical property of the lawnmower.
    – causative
    Apr 21 at 7:36
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    I'm quite unhappy by your "paper model theory": There is a big difference between e.g. a e.g. a paper model of a set of triangles which cannot be assembled to a closed body; and an equation that happens to be written on paper. Nothing keeps us from writing a wrong equation. This paper model is symbolic, not material; it is information, not physicality. It is used to store and transmit a symbolic model in a computer or our head. The philosophical question here is whether such models can be a reason to say "math describes reality in our heads". Apr 21 at 8:13
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    @causative The example is telling because obviously the Rubik's cube is a physical model of some algebraic system (or rather: The other way around?) which prevents errors. It is not symbolic! By contrast, the Sudoku or equation on paper is clearly not such a model: Nothing prevents you from writing down wrong numbers. The rules are in your mind only. Apr 21 at 8:41
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    @Peter-ReinstateMonica Well, nothing stops you from painting the Rubik's cube in the solved configuration. That you don't do that is a rule "in your mind" as well. But it is still a property of the Rubik's cube that it can be solved, or can't be solved, without doing things like that. Just as it is a property of the Sudoku that it can be solved, or can't be solved, without doing things like that.
    – causative
    Apr 21 at 8:50
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See Quine's Naturalism: according to Quine, the ontological questions are decided on pragmatic grounds, whose standards do not differ from the evidential rules of science itself. In a nutshell, for Quine philosophical questions in the end are converted into scientific ones.

Thus, also mathematics and logic itself must be revised if necessary, in order to accomodate new facts discovered by empirical sciences.

See also Naturalism in the Philosophy of Mathematics and Penelope Maddy, Naturalism in Mathematics (1997).

Also relevant the so-called Putnam-Dummett debate: "Is logic empirical?"

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  • I need to do more hard thinking (and then reading) to try to frame the question in entirely new terms. When we say “empirical” simply means “as a result of observation”, on the one hand, it implies “observation” or perception is synonymous with a kind of perfect, definitive knowledge; that there is no greater form of certainty than seeing a cat in front of me. This is like Descartes, that whereas there may always be an inherent uncertainty or improbability in a statement like 2+2=4, if we denied the existence of percepts (like a cat), it would seem to be a universal denial of everything. Apr 22 at 13:49
  • On the other hand, “observation” immediately calls into question the nature of being a self, a perceiving subject, an experiencing entity in time. If I “perceive” my own thoughts, do we start to unravel the self, where there wouldn’t really be a “perceiver” of one’s thoughts, since the “thinker is the thought”. And even if I can perceive my own thoughts, is the “empirical” component that I use (analytic) reason to apply empirically learned laws of the world? If we take “empiricism” away, we ask, “How do you know what laws are true?” If take “analyticity” away, “how do you apply rules?” Apr 22 at 13:54
  • The OP should note that Quine, while one of the best mathematical empiricists is not the first. John Stuart Mill is considered by secondary sources to be an empiricist given his version of radical empiricism. Hartry field is another figure that should be explored in the question of just how empirical is mathematics. I'd get a copy of Linnebo's work: google.com/books/edition/Philosophy_of_Mathematics/… I could outline other contemporary accounts that reject Plato's thinking as superstitious, but this book will give you the lay of the land.
    – J D
    Apr 22 at 14:00
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Yes. Mathematical laws are empirical, if we consider a universe not just a set of physical laws but a flow of qualia.

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.

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  • You speak as if there's one set of mathematical laws, but there are many. Something that's a paradox under one set of laws, such as classical logic, may not be under another set of laws, such as intuitionistic logic.
    – cjs
    Apr 22 at 7:55
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Most mathematicians are dead mathematicians — at least the ones over 100 years old!!

Now this is strange because most mathematicians also believe that they have discovered math truths. That is they believe that their truths have always been true; they just happened to discover them.

So it's a peculiar fact that mathematicians are finite contingent beings who have a penchant for forgetting their umbrellas, yet give birth to eternal unchanging 'babies'.

This is strikingly different from most else:

Plato in Meno shows that we learn through recollection — anamnesis. Socrates makes a slave boy of Meno who had no knowledge of geometry correctly solve a problem of geometry. The point of Socrates (Plato) is that that recollection is incited by Socrates but it is a recollection of something the boy already knows 'in his soul'. The slow unfolding of that recollection happens of course in time but it is of a truth that is timeless.

In other words math is Platonic, mathing is empiric.

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Mathematics is indeed NOT empirical, what IS empirical though, is the application of Mathematics in a specific field or problem.

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No.

To complement Jo Wehler's good answer, a mathematical truth is not empirical, it is purely logical.

This is because in principle, a brain in a jar with no sensory information about the outside world could deduce any mathematical truth by careful application of logic alone, given a set of axioms.

It would be impossible for the brain to figure out any empirical truth precisely because they require information about the world from which the brain is cut off. There are no empirical proofs independent of information about the outside world.

One could argue that the brain must have gotten the axioms from the outside world, but even if it did the axioms are not information about the outside world. Just rules of a game.

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"Empirically true" might mean that something may or may not be always true but the current observations show it to be true.

"Mathematically true" is something that can be proven to be always true under certain axioms and ways of deduction.

Everything that is mathematically true is also empirically true. If it isn't then the used mathematical model does not match the world in which empirical facts are gathered.

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    Apr 22 at 13:23
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Wikipedia says:

Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure.
...
The term empirical comes from Greek ἐμπειρία empeiría, i.e. 'experience'. In this context, it is usually understood as what is observable, in contrast to unobservable or theoretical objects.

When considering something like Cantor's result that there are more real numbers R than natural numbers N, or the difference between countably infinite and uncountably infinite sets, there are no objects or sets of objects in the universe that correspond to these notions. (Even if the universe has an infinite number of objects in it, that is still a countably infinite set, and thus smaller than any uncountably infinite set.) Nor can we sense or experience these things. They are just ideas in our heads.

I suppose there are alternate definitions of the word "empirical" that could embrace ideas that are just in our heads. But, as you point out, going much down that path basically makes the world "empirical" meaningless, there's nothing that can be distinguished as "not empirical."

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