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When someone experiences the mental clarity of 2 + 2 = 4, is this a form of experience similar to let's say, seeing red, or the sour taste of a pickle.

On the one hand it seems like it is a form of experience, just a mental one. On the other hand, Math has a tremendous distinction, in that it seemingly coerces the result from the one thinking as opposed to other experiences where the subjective and the "objective", both have an effect on the subjective experience. (e.g. the retina of the subjective changes the experience of light).

On the same note, one can argue that the mind of the subjective thinker forces them to understand that 2 + 2 = 4, as an animal has no concept of this; therefore one can argue that Math is just as epistemological as seeing red.

I believe Immanuel Kant talks about this, are there good arguments for or against the epistemological roots of Math?

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    It is certainly a form of experience, but people would rather say involving mathematics rather than being mathematics. It is quite dissimilar to visual and other perceptions though, whereas they are purely qualitative this one involves cognitive content (number, equality, etc., concepts) and a judgment or "sense" of objectivity ("clarity"). Kant did talk about this, but he focused exactly on non-experiential aspects of intuitive construction, and he would reject the idea that such personal psychological aspects are salient to the nature of mathematics.
    – Conifold
    Feb 7, 2022 at 21:05
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    Learning and understanding math is certainly a type of a posterior mental experience, but math as an independent entity qua itself may not be anyone's experience at least for Platonists or other math realists per Kant's famous synthetic a priori designation of math. Feb 8, 2022 at 4:52
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    Maybe relevant The Epistemology of Visual Thinking in Mathematics: mathematical experience needs "symbols": drawings (ancient Greece math) and signs (algebra, from Renaissance onward). Feb 8, 2022 at 8:30
  • Thanks, @MauroALLEGRANZA. That's a fascinating article!
    – J D
    Apr 21, 2022 at 19:12

4 Answers 4

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You're wrong about animals not having a concept of "2+2 = 4". Since 2009, it has been known that even young chicks can not only count but also add and subtract small numbers (1, 2, 3). This should not be surprising; humans are just animals with a larger mental capacity after all.

And if you know a bit of mathematical logic, you would also know that there is no such thing as a single "mathematics". "Mathematical platonism" is often brought up, but it is in fact an ill-defined concept, and you'll only understand this if you know basic FOL (first-order logic) and know various foundational systems for mathematics (with all the technical details), such as PA, ACA0, ATR0, Z2, ZFC, ... (See here for where to start learning all this, and here for a brief sketch of the hierarchy of common foundational systems.)

The correct answer to your question is that only a miniscule fragment of mathematics, somewhere between ACA and Z2, is relevant to the real-world. PA− axiomatizes basic counting, and PA (which is PA− plus induction) extends that to support reasoning about basic counting, and so far we have good empirical evidence that theorems of PA are true when translated into real-world statements. This empirical evidence extends to ACA via a syntactic interpretation of the 'sets' in ACA (see here for more details), but fails to extend much further. I think on conceptual grounds one can justify up to ATR0, but beyond that it is unclear what 'sets' would mean in the real-world. Logicians generally do not believe that there is a real-world interpretation for anything from Z2 onwards.

So one can expect that ACA has platonic meaning, but higher mathematical reasoning that is not support by weaker subsystems of Z2 cannot be justified (as of now) to have platonic meaning, so we are left with ascribing them with more formalist meaning.

"2+2 = 4" is easily observed in the real world, so you should expect any conscious being with a small amount of mental capacity for forming and acting upon empirical hypotheses to realize this fact. There is no reason to think there is any "coercion" in this realization. Also, mathematics is a socio-historical construct, so mathematical concepts beyond PA− cannot be blindly considered on par with "2+2 = 4". In fact, numerous people who followed their mathematical desires ended up proving nonsense.

Maybe you need to reconsider what "2+2 = 4" really means in the real-world. It implies statements like "If you (using a 2 L jug) pour 2 L of water into a pail, and then pour another 2 L of water into the same pail, you will get the same result as if you (using a 4 L jug) pour 4 L of water at one go into the pail.". The reproducibility of verification of this statement is what makes it an empirically verified instance of the abstract statement "2+2 = 4". If this instance is not objective, then almost nothing else is, and all science is indistinguishable from magic.

But your question title has the nebulous term "mathematics". As I said above, what mathematicians count as "mathematics" goes far beyond "2+2 = 4", so your title's question cannot be answered if you don't even specify which foundational system for mathematics you are considering. One can defensibly argue that all mathematics supported by ACA are discovered in the sense of translating to a discovery about the real world, but general theorems of ZFC are merely discovered as symbolic features of ZFC that may not have real-world relevance (and hence cannot be experienced).

Note that even for PA, you cannot really experience certain theorems, such as a suitable translation of the following:

  For every program P and input X, either P halts on X or P does not halt on X.

The problem is that you cannot experience "non-halting" in any meaningful way unless someone can concretely justify it to you (say via game semantics). And by the incompleteness theorems one cannot always determing halting behaviour, not to say justify non-halting even when it holds.

However, we can experience every instance of "halting" (by following the program execution until it finishes), and in some sense that is really the limit of what part of mathematics we can experience fully. After all, every theorem that is proven in a foundational system is witnessed by the fact that the proof verifier for that system halts on the purported proof of that theorem. But this is only an experience of the formal system, and not of any 'truth' stated by the theorem. Even a simple claim like "∀k∈ℕ ∃m∈ℕ ( k = 2·m ∨ k = 2·m+1 )" cannot be experienced fully; via game semantics you can experience "∃m∈ℕ ( k = 2·m ∨ k = 2·m+1 )" for more and more instances of k∈ℕ, but only finitely many of them.

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  • In case the "2+2 = 4" example is too simple, consider the more interesting example "∀k,m,n∈ℕ ( (k·m)·n = k·(m·n) )" where multiplication is interpreted as school-book multiplication in base 10! It yields instances like "(9·127)·9721 = 9·(127·9721)". Why on earth should doing the school-book multiplication in a different order yield the same result? Clearly, it is due to some underlying real-world fact. What do you experience when you finish the multiplication (on paper) and find the same answer?
    – user21820
    Apr 21, 2022 at 17:17
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    You provide a nice counter-example to illustrate that mathematical results are not confirmed by an experience of clarity. - Instead, the correctness of the associative law of multiplication has to be proved, possibly by induction. - One can do the proof in a closed room without any reference to the outside word. One just uses the basic axioms of arithmetic and the rules of logic. - For me this fact indicates that the mathematical law in question does not derive from some underlying real-world fact.
    – Jo Wehler
    Apr 21, 2022 at 19:41
  • @user21820 Just don't try that in the octonions! en.wikipedia.org/wiki/Octonion
    – user4894
    Apr 22, 2022 at 0:06
  • @JoWehler: That's actually incorrect. If you actually follow the link I provided for PA−, you will see that "∀k,m,n∈ℕ ( (k·m)·n = k·(m·n) )" is an axiom. Why? Any algebraist will tell you that this is a discrete ordered semi-ring. And any logician who studies PA will tell you that PA− is the correct base theory, not the one you are talking about. The equivalence between the one in your mind and the one referred to here (i.e. PA− plus induction) is misleading; it fails to capture the 'true' structure of arithmetic.
    – user21820
    Apr 22, 2022 at 9:27
  • The fact that logicians all feel that PA− is the correct base theory to capture the basic facts of real-world counting, shows that your conclusion is wrong. They do not choose those axioms whimsically; they chose the semi-ring axioms because it is exactly what the real-world phenomena obey! If nothing in the real world counting has structure described by PA−, nobody would come up with PA−!
    – user21820
    Apr 22, 2022 at 9:30
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Answer

According to some philosophers of mathematics, yes. Such thinking is known as mathematical empiricism, and tends to be a highly empirical view of mathematics which is a minority position among actual mathematicians. It supports psychologism more broadly.

As far as the advantages of a philosophy of mathematics that appeals to its empirical qualities, it would both:

  1. Be a strong ally of scientific and mathematical constructivism where mathematical truths are internal language productions in response to experiences.
  2. Also be amenable to be integrated into a bigger picture of anti-realist empiricism such as van Frassen's theory of constructive empiricism (SEP).

The important thing to take away from constructivist and empirical notions of mathematical philosophy (besides being they are unpopular with actual mathematicians according to Linnebo) is that they stand strongly against Platonic mathematical thinking (SEP) as a foundation for understanding mathematics.

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1.) I agree about the mental clarity of 2+2=4. But who has a mental clarity about the result of

1.537 + 484=??

The example illustrates that mental clarity does not prove the correctness of the result. Mental clarity of 2+2=4 results from our learning the multiplication table which took some time in school. To obtain the correct result of 1.537 + 484 one has to apply the standard algorithm of addition, which one learns some years later in school.

2.) Mathematics is not a form of experience. We do not experience the correctness of mathematical propositions, we prove their correctness on the basis of the correponding axioms and following the rules of logic.

Hence finding and proving mathematical theorems cannot be compared to seeing redness in any epistemic sense.

3.) Kant expresses his position about mathematics in Prolegomena to Any Future Metaphysics: Mathematics constructs its concepts in pure intuition.

That’s a difficult passage, but fundamental for Kant’s philosophy. Moreover, today severe objections are raised against Kant’s understanding of mathematics as a science which provides synthetic knowledge a priori.

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  • I find 2. an interesting claim. Is not the cognitive dissonance of a mathematician in response to a logical contradiction in a proof an experience, and if so, why not?
    – J D
    Apr 21, 2022 at 19:10
  • @J D Each indirect proof in mathematics aims at a logical contradiction between the assumption made in the beginning of the proof and some conclusion of the assumption. The corresponding experience is not a cognitive dissonance but satisfaction about having reached the goal. - I do not understand what your point is. Could you please elaborate a bit more?
    – Jo Wehler
    Apr 21, 2022 at 19:30
  • Sorry I wasn't more explicit, Jo. So, in that view both the cognitive dissonance, an emotional feeling akin to anxiety, and satisfaction, a feeling akin to pleasure are key psychological experiences in the proof. To that, add the visual experience of geometric proof, and the linguistic experience of analysis, how isn't it such that mathematics is largely done by and for visual, linguistic, and emotional experience? I have no problem that you draw a box around a deductive proof in analytic geometry and say, look at the scribbles! I'm just curious why all else is excluded as mathematics?
    – J D
    Apr 21, 2022 at 21:18
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Compare, "I experienced addition for the first time today," vs., "I performed the operation of addition/used the addition operator for the first time today." You might go on to say, "I experienced myself performing the operation of addition today," and go on to even more unwieldy sayings ("I performed the operation of my own experience of my first use of the addition operator today," say). The two base cases each sound 'right' in their own context; the first might be more indirect (lending itself to the hideous iterates) but still sensible from e.g. a literary perspective (you could do well to convey a character's unique persona by having them describe all their conscious states in experiential terms).

That there is, however, a knowledge-theoretic distinction between passively experiencing a conscious state and operationally bringing it about, is indicated abovewise. For Kant [not officially, i.e. not as he outright defines apriority in the beginning of the first Critique, but indeed as he makes the point in the Groundwork], apriority involves "spontaneity" in the causation-like sense. This indication runs out of some clarity when we factor in the existence of action theories of perception, but one way to salvage the notion (and Kant's way out regardless) is to implicitly have a third epistemic option on this spectrum, i.e. interactive knowledge (as opposed to only passive, or only proactive).

Then there is some interactivity to even usual counting, even when we abstract to larger cases of addition and line up/add up/cross out/resummarize lists of digits (the a priori side of mathematics interacts with the empirical presence of the digit symbols and their lists, seeing as the symbols themselves are proactively caused, after all).

The feeling of clarity is a matter over and above the above, though. Again working from behind the Kantian scenes, I would offer the suggestion that this is, like respect, an a priori feeling. Albeit Kant clearly says that intellectual pleasure is not, just for its intellectualism, morally superior to empirical pleasure, still, if there is a non-pleasurable but still positive feeling involved in intellectual certainty, I would be hard pressed to parse out its apriority, then. Maybe it isn't, just for its apriority, also morally infused like respect is, but if the premises of the system otherwise compelled that inference, well, oh well. Physically, I would like to know if positive sensation/affect associated with more [so-called] 'carnal'/'fleshly'/'worldly' desires involves the same neurons firing, or the same neurochemical mediation, as in the case of positive sensations/affect associated with puzzle-solving, making conceptual connections [e.g. wordplay], etc.

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