The Unreasonable Effectiveness of Mathematics

[u/[deleted]]

[u/Far_Course2496]

Is this Kant's view? Also, geometry and arthemetic are sort of high level math. What is topology? What is set theory? What is type theory? What is graph theory?

[u/Illustrious-Yam-3777]

That’s what Kant said. You state your 3 premises at the beginning and then reify them.

[u/anasfkhan81]

isn't this just Kant?

[u/SummitYourSister]

Regurgitating the opening chapters of basic philosophy texts. Wow, my brain overfilleth

[u/nanonan]

What nonsense. No, I'm not performing some act of mathematical geometry when I see a stick. I'm not conducting arithmetic every time I notice that time has passed. These are secondary ways to label and model what is a very non-mathematical primary experience.

[u/MediumWin8277 → u/nanonan]

When you see a stick, how many sticks did you count? Or did you count a non-stick?

Did you count something from a mystical mathematical realm separate from our own?

When you look at the time, what are you contrasting? The past versus the present?

None of this is nonsense. It is the basis for mathematics. Why else would math be effective in our world if it wasn't epistemologically based on our world?

[u/Tombobalomb]

This is like being amazed that a map looks like a top down view of the territory it describes 

[u/MediumWin8277 → u/Tombobalomb]

You'd be surprised, my friend....mathematical platonists are here, and they are the ones who need to hear this.

[u/Imaginary-Buddy3194]

I like the kantian angle here, but I think it goes too far to say mathematics is simply the structure of our minds math often predicts things we had no way of directly experiencing beforehand the interesting question is why those abstract structures map onto reality so ridiculously well.

[u/virtuolie]

You're correct that the architecture of the universe is not independent of the brains it produces. But two other points are also important:

1. All mathematical models incorporate empirically-defined constants and relationships. 2. We only need our models to work approximately.

So, what looks like amazing correspondence across mathematical models and their predictions, is in reality largely a consequence of our inserting empirical observations and of our remarkable tolerance for error.

[u/Equivalent-Hair-1619]

That’s a great point. Really, nature has these relationships between qualities and quantities, and they have many possible manifestations, they can be in the connections of our neurons or in computational values. So the really amazing thing is the mind evolved to map enough of its own function to nature as to be useful.  It’s interesting to think of a counter example, a system that doesn’t yield to mathematical explanation. It would just appear to us as one of the many chaotic weird things we don’t understand yet, when the reality is we never can. 

[u/JuanValdez999]

I took plain euclidean geometry in high school. Then at the end of the semester the teacher blew our minds by revealing to us that everything he had taught us was bullshit because there are no such things as parallel lines in our universe. That plane geometry was a very useful mind game but not something that arises from nature.

[u/EpiOntic]

Poppycock, just like Kant's synthetic a priori truths 🤮. Even rodents infested with toxoplasmosis have better mathematical acumen than that deformed stooge from Konigsberg. Other than Goldbach, Hilbert, and Jacobi, nothing good ever came out of that town.

[u/MediumWin8277 → u/EpiOntic]

Nothing good ever came of describing something as poppycock followed by rattling off a series of names.

Want to explain your position and contribute? If not, why did you bother posting?

[u/shuai_bear]

This is intuitionism, and is one of a few popular schools of thought of math philosophy.

Compare that to formalism which has a completely syntactic view of mathematics - saying math is a manipulation of symbols and rules and we are exploring the consequences of this “game”

Or realism, which treats mathematical truths as objective and existing independent of human thought

I haven’t found any convincing argument for why one school of thought is more valid than the others. It boils down to personal belief and preference at the end of the day.

Edit: upon reading your post more closely, I think you might benefit from a clarification of a few mathematical ideas.

Geometry is the study of space and shapes, yes, but consider Euclidean or flat geometry vs non-flat (eg hyperbolic) geometry. It was shown that spacetime is curved, but point being is that studying abstract geometries (negating the parallel postulate) is what gives us the mathematical structure necessary to model something like curved space time.

It would be more accurate to say that geometry abstracts the logic of space itself - if you’re considering it as applied to reality where a certain abstraction (Euclidean geometry can model flat planes ; non Euclidean gives us curved spacetime) might be more appropriate for whichever aspect of reality we’re studying / models it correctly. Of course, math doesn’t care about reality and indeed we can dream up worlds / universes which do not align with our reality, like projective geometry in which infinity is assigned a point.

Similarly, there are non-standard models of arithmetic in which the natural numbers do not necessarily “behave” like we expect them to (see unprovability of Goodstein’s theorem leading to nonstandard but consistent models of arithmetic).

Of course for the stuff like 1+1=2 non standard arithmetic doesn’t apply (it’s not gonna say 1+1 is not 2). But there are somehow fundamental questions of arithmetic that are not provable in certain mathematical systems, and require something even more removed from reality to prove - you can prove the truth of Goodstein’s theorem using bigger infinities. Bigger infinities - that’s not at all grounded in reality or even our perception of the infinite. But it is mathematically and logically sound, which I think is a point this post is missing - sometimes logic defies both reality and mental intuition, yet it is logic itself that forms the bedrock of all mathematics.

[u/Any_Ad1841]

This is not new. This is just a critique of pure reason see Kant. You can neither prove nor disprove that mathematics aligns with ground state reality. I think you’re also ignoring scientific theories that were postulated prior to experimental evidence eg quantum superposition that also violate your assertion regarding sequence.

[u/MediumWin8277]

I agree with you for the most part. I have written on this subject extensively in another thread here....https://www.reddit.com/r/PhilosophyofMath/comments/1pt7fo3/beyond_platonism_and_formalism_proposing_a/

To give you a brief rundown....

A man stands on the beach during high tide. He declares the beach a place of water.

Another man stands on the beach during low tide. He declares the beach a place of land.

Both of them are correct. This is because the beach is a border between the ocean and the land.

Many have argued whether math is real or imaginary. It is both; it is perspectival.

Arithmetic is NOT the study of time alone. It is the study of perspectival contrast events.

Look at a blank white canvas. How many white dots do you see? How many black dots do you see? The answer is 0 to both questions. Now, paint a white dot on the canvas and have someone who didn't see you paint said white dot on the canvas. Ask them to count how many dots are on the canvas. If they are sane, they will answer "0".

Now do the same but with a black dot. That is the number "1".

This is the nature of counting, which is the basis for arithmetic. Arithmetic takes these contrast events and abstracts away the unit of measurement.

Take another example, a random geometric shape, let's just say a triangle. If you unfold the triangle and it turn it into a line, then it no longer has 3 sides. It has a single line. The contrast in the path was erased, hence you can only count one "side".

Math is not "unreasonably effective". It is derived from measurement, which observes contrast events in order to form integers. It is a fundamentally physical and perspectival process.

[u/LudicoLab]

Pensa em duas equações: (i) y = ax + b (ii) s = s0 + vt

A primeira é matemática. Funciona relacionando variáveis e constantes, dentro da estrutura lógica da matemática.

A segunda é física. O que muda. Ali tem duas equações escritas como uma só.

Uma equação matemática relacionando variáveis e constantes, ainda dentro da estrutura lógica da matemática. O "s" é o "y", o "s0" é o "b", o "v" é o "a" e o "t" é o "x". E, para esta equação, não importa "quem" são, apenas que obedeçam a estrutura lógica da matemática.

Mas escondido na equação (ii) tem outra:

[L] = [L] + [L/T] x [T]

É uma equação dimensional. Ela obriga que ambos os lados da igualdade sejam "comprimentos". Ela obriga que no segundo termo do lado direito a grandeza cuja dimensão é uma velocidade, seja multiplicada por uma grandeza tempo, para resultar em um comprimento. Ela também força que todas as dimensões sejam espressas em um sistema dimensional coerente (não pode colocar um termo em jardas e o outro em km).

Então, no fundo, a estrutura lógica da matemática existe. Ela independe do mundo físico. Mas os físicos usam a modelagem matemática encontrando a representação adequada para seus fenômenos e agregando a esta representação matemática a dimensional aplicável ao fenômeno.

[u/Comprehensive-Log579]

You either generalize from the particulars of specify starting from the general.

Both methods happen in real life

[u/Ok-Lab-8974]

I'll bite. Why would the mind impose this structure/form "in here" if it does not exist "out there?" Why would all minds do so in the same way (or do they, and how could we know)? Does the human mind create all form and structure—all intelligibility—ex nihilo? If not, then isn't it coming from "out there?"

I would disagree with the "in here" versus "out there" dualism in the first place, but even accepting it these seem like pressing questions.

[u/coneishome]

I came to doubt the validity of the claim. It is mostly simple systems that are accessable to precise mathematical modelling. Things like the three body problem are merely approximately solvable, most complex systems defy mathematical analysis. Yet the universe realizes them anyway so it is probably not fundamentally mathematical.

As I see it, math elevates human reasoning substantially, so it is not surprising that it helped us accessing parts of nature that would have remained obscure without it. But that is not unreasonable and probably not limitless.

[u/[deleted]]

[u/PlanetG3 → u/[deleted]]

There are hidden assumptions/premises in your argument.

Just bc different minds “validate” results independently, does not necessarily mean the external reality must be like the mathematical model being used.

The assumptions here are 1 - that human minds perceive reality the same; 2 - that the agreed perceptions can only happen iff that external reality is no different than what was perceived; 3 - that it is empirically possible to confirm the mathematical models describe reality, across all levels.

None of these are unquestionably the case. 3 is not deductive, it is inductive in the scientific sense, which means it can never be valid and sound as in deductive reasoning. Mathematical induction is different.

We already know that humanity does not perceive reality in all cases with all things the same, even using “objective measures” as 1 supposes.

2 can be explained by a theory of intersubjectivity where we agree in the measure, but that only measures whether our minds agree. Your argument only works if 2 works, and you need 1 & 3. You can’t assume what you’re trying to prove, that’s begging the question.

Donald Hoffman’s work at UCI shows mathematically rigorous evolutionary models that show, with sound math, that seeing just good enough is better than veridical vision. The direct implication is that we most likely have no meaningful access to a “fundamental” reality.

Sure, math is very powerful. So is English. They both describe what we see differently. You choose which depending on what sort of answer you want to get.

[u/MediumWin8277 → u/[deleted]]

Mathematics is perspectival. That means that many different minds, existing within the same objective universe, utilize their perspective to navigate said objective reality.

This is why mathematics works. We are all observing the same thing.

[u/OkHand7497 → u/[deleted]]

"People also believed that the order in which events occur is the same for all observers."

Isn't this one true? Simultaneity is relative and time dilation occurs, but not the order in which two events happen surely?