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Is mathematics a mental idea?

According to this answer, a mental idea cannot exist without a mind.

If mathematics is a mental idea, what does this imply about the laws of physics which can be modeled mathematically? Is there a way to fully explain this without resorting to a conscious mind behind them (either within us or perhaps outside us)?

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    Mathematics is a human activity, a language, a practice. – Mauro ALLEGRANZA Sep 4 '19 at 6:02
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    What does it mean "mathematics itself" ? Do you suggest that there is also a mathematics that "other from" mathematics ? – Mauro ALLEGRANZA Sep 4 '19 at 6:32
  • @MauroALLEGRANZA but is math a mental idea? – michael Sep 4 '19 at 7:15
  • Kant thought so (more or less), according to him it means that we discover laws of physics because our minds ("understanding") organize our experience according to those laws in the first place:"If he is to know anything with a priori certainty he must not ascribe to the figure any thing save what necessarily follows from what he has himself set into it in accordance with his concept." – Conifold Sep 4 '19 at 9:06
  • Everything object is an idea and if it weren't we wouldn't know about it. The question is whether some ideas represent objects that are independent of mind. The mental status of mathematics allows us to speculate there are no such objects but does not rule out the possibility. If it did then all philosophers would be idealists of some sort. – user20253 Sep 4 '19 at 17:08
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One approach to mathematics as a mental idea is intuitionism. Here is Rosalie Iemhoff's description of it:

Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds.

Here are the questions:

If math is a mental idea, what does this imply about the laws of physics which can be modeled mathematically? Is there a way to fully explain this without resorting to a conscious mind behind them (either within us or more likely, outside us)?

It is unlikely that one can fully explain anything. One offers the best explanation one can and then one remains open to evidence countering the explanation. There may be a conscious mind behind everything. This would be a form of panpsychism, pantheism or theism.

Given that mathematics, from an intuitionistic perspective, is a mental idea it does not mean that whatever the laws of physics are supposed to refer to is not actually out there, but it does suggest that the mathematical formulations of those laws are also mental ideas communicated between physicists which can be falsified with new evidence.


Iemhoff, Rosalie, "Intuitionism in the Philosophy of Mathematics", The Stanford Encyclopedia of Philosophy (Summer 2019 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/sum2019/entries/intuitionism/.

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  • Inside, outside, and containment in general is a conceptual metaphor, and might be misleading. en.wikipedia.org/wiki/Conceptual_metaphor – J D Sep 4 '19 at 15:50
  • This is not really the distinction. Platonism and various other idealisms also depict mathematics as generated by the mind, or at least as part of it, but they put the mind at the root of reality. Intuitionism admits that mind is uniquely human (or animal), and not that of God or some other underlying mental reality. – user9166 Sep 24 '19 at 17:21
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Mathematics is language. All languages exist on verbal levels of our intellect.

Laws of physics exist independantly of mathematics or any other form of language. Laws of physics are observable and can be expressed mathematically.


EDIT 16, october 2019.

Mathematics, like any language, is a system of concepts and relationships between them. All concepts exist on a mental level.

For example, you can clearly perceive with your eyes the size of two circles on the picture. Your eyes see the size of circles directly, even before you express it in numbers (by using some mathematical tool).

The physical laws can be described very accurately with mathematics. However, direct perception is more advanced than the language. You can clearly see the physical phenomenon without the need for any descriptions. The conclusion is that knowledge comes from direct perception.

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  • how does this answer the question? thanks – michael Sep 24 '19 at 5:13
  • It depends on whether by 'laws of physics' you are citing the symbol "laws of physics" or the referent, laws of physics. :D en.wikipedia.org/wiki/The_Meaning_of_Meaning – J D Sep 24 '19 at 15:34
  • I'm refering to what we call "the laws of physics". The referent or the observable phenomena of material world. Not the map, but the theritory. – Marino Klisovich Sep 24 '19 at 18:01
  • Mathematics is not language. This is at best an analogy. – M. le Fou Oct 25 '19 at 12:29
  • @M.leFou — I'd reconsider that, if I were you. Mathematics is clearly a system of symbolic representation, which means that it is (at least in part) a language. – Ted Wrigley Nov 15 '19 at 15:11
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Lovely question Michael.

If that @RodolfoAP answer is literally exactly true,

  • science is based on math
  • math is based on perceiving mind
  • my perception (mind) is all = solipsism

Since all modern life is science-based solipsism is the ultimate philosophy.

As a secondary proof of

Anything is true if we will it strongly enough (The Secret!) consider:

  1. That answer is the highest voted atm
  2. That answer is the wrongest in that it directly contradicts one of the cornerstones of 20th century logic — Tarski's undefinability of truth (subsumes Godel incompleteness )

And if you don't take the math-defines-truth bit too literally and only focus on "It's an ill-formed question" (others there like @celtschk do it in more detail in other answers), this may not give quite as egregious results as math-defines-truth but it is still nonsense as @conifold points out in funny contradictions of LPists/analytic philosophers

Personally I find the next most-voted answer by @JD more insidiously dangerous than RodolfoAP's. I won't explain that further than to state without more "proof"

  • I am a – maybe unwilling – platonist
  • Plato would be mighty offended with 20th century (cantorean/hilbertean) "platonism"

The best answer to this was given by @Jobermark explanation and defence of intuitionism. Perhaps here.
Seems to be deleted!

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    I have no idea what this means. – niels nielsen Sep 4 '19 at 7:57
  • @nielsnielsen Nor do I about your comment. The difference is that my answer has a dozen paras a couple dozen sentences. Your comment has just one! – Rusi-packing-up Sep 4 '19 at 8:02
  • Would it help @nielsnielsen if I said that extremity of math-is-analytic (RodolfoAP) as well as extremity of the other side : msth-is-synthetic (JD answer) produce nonsensical results? – Rusi-packing-up Sep 4 '19 at 8:13
  • is that teh only option though. could there be a higher mind over everything as max planck thought? – michael Sep 4 '19 at 11:14
  • no, it would not help. I think I'm done here. – niels nielsen Sep 4 '19 at 17:27
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Long comment

Idea is a complex philosophical term.

According to Descartes’ Theory of Ideas

the mind is an existing substance, and thought or thinking is its attribute. An idea is a mode of thinking. In being a mode of thinking, an idea is understood as a way of being (an instance of) thinking, or an idea is way in which an instance of thinking is manifested. This is similar to what Descartes says about a body : [t]he nature of a body is to be extended (in length, breadth, and depth). A body is a substance, and extension is its attribute.

ideas are cast as modes of thinking that represent (or present or exhibit) objects to the mind — objects such as a man, or Pegasus, or the sky, or an angel, or God. [...] Descartes is careful to not identify ideas as pictures or as visual images, but instead says that they are as it were images of things.

Descartes' point of view can be compared with modern concept of Mental Representation :

[this theory] takes as its starting point commonsense mental states, such as thoughts, beliefs, desires, perceptions and imagings. Such states are said to ave “intentionality” — they are about or refer to things.

Is this what yo mean ?

Maybe your are referring to mathematical concepts, like e.g. that of number or set. In this case, the question will be : the concept (idea) of number is an idea (representation) of what ?

Or instead are you alluding to a "subjective" approach to mathematical concepts ? In this case, they are construction of human mind (see Kant's theory of the construction of mathematical concepts and Brouwer's Intuitionism.

More details are needed in order to go on with the discussion...

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  • i just find it curious that the laws of physics can be modeled mathematically. doesnt that imply some sort of mind behind them as explained in the question – michael Sep 4 '19 at 11:12
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    @Michael - as per many many posts on this site, the question is not "curious" : it is a philosophical problem unsolved since (at least) Plato. Many many different views have been proposed and discussed; in a nutshell, the "big divide" is : if math is discovered, how we "perceive"/learn abstract entities like numbers and sets ? If math is invented by human mind (like e.g. chess game) how is it possible that it "works" so well ? – Mauro ALLEGRANZA Sep 4 '19 at 11:22
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Is mathematics a mental idea?

Mathematics is an abstraction, an idea, a thought, a meaning and is not physical.

According to this answer, a mental idea cannot exist without a mind.

It depends on who you ask. Descartes would disagree with his dualism. Daniel Dennett would disagree with Descartes. Theology still tends towards dualism since it preserves supernaturalism, while cognitive science largely rejects it.

If mathematics is a mental idea, what does this imply about the laws of physics which can be modeled mathematically? Is there a way to fully explain this without resorting to a conscious mind behind them (either within us or more likely, outside us)?

It implies that consciousness generates meaningful expressions in mathematics and science and uses linguistic elements like syntax and orthography to describe, explain, and predict physical phenomena. Consciousness seems to be required for theory generation. Ever read a theory determined and expressed by an unconscious mind?

These are generally accepted ideas among contemporary philosophers of mind. Where they disagree is on meaning, terminology, and methods when confronting problems like the hard problem of consciousness.

This evokes metaphysical positions such as naive realism, transcendental idealism, and philosophical realism, etc.


EDIT 2019-09-24

If mathematics is a mental idea, what does this imply about the laws of physics which can be modeled mathematically? Is there a way to fully explain this without resorting to a conscious mind behind them (either within us or perhaps outside us)?

It depends on your metaphysical presupposition. If you believe in physicalism for instance, especially a non-reductive form, you according to Jaegwon Kim, you accept three principles at a minimum:

  1. The mental supervenes on the physical.
  2. The mind-body duality is a category mistake.
  3. The mental is wholly dependent and characterized by the physical.

(See Ch.1, Philosophy of the Mind)

As such, the "laws of physics" is both a symbol that given the triangle of reference refers to patterns in observation of phenomena that correspond with patterns in noumena, and das Ding an sich presuming a dichotomy between the observer and the external universe.

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  • what does dualism have to do with whether an idea can exist without a mind? – michael Sep 5 '19 at 10:07
  • @michael I'm not a dualist, so maybe I misunderstand the position. I guess I would say that Cartesian duality has a distinction between the body and the mind, but if one can divorce the body from the mind, one can divorce ideas from mind along the lines of Platonic thinking. If the mind supervenes on the body, then ideas necessarily must be tied to the mind? – J D Sep 5 '19 at 12:42
  • @michael I updated my answer to better answer your question – J D Sep 24 '19 at 15:52
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This is a knotty problem and there are many different perspectives on it but I would recommend looking at the details.

You might need a mind to discover the laws of physics and program a computer to simulate them. But when it's running it just carries on by itself and does not require any mental intervention. So there's nothing here to imply that the universe needs a mind in order to exist or go about its business.

It might require minds to make sense of it (but we already knew that).

You can make a case that math is "all in the mind" but if so it would be unusual to think that it was necessary for physics (but maybe necessary for some kind of human understanding of it).

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  • Welcome to SE Philosophy, and thanks for contributing with a question/response! If you haven't done so, please take a quick moment to take the tour. philosophy.stackexchange.com/tour More specifics can be found in the help center. philosophy.stackexchange.com/help – J D Nov 15 '19 at 22:44
  • "But when it's running it just carries on by itself and does not require any mental intervention" - anything to back this? – michael Nov 16 '19 at 21:03
  • I've written physical simulations on computers before and they ran fine when I turned my back on them. This is all I'm claiming. For all I know the actual universe does require the constant oversight of a conscious deity but we haven't established that with this particular argument. – user68014 Nov 16 '19 at 21:40
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I feel like I repeat this point a lot, but so be it...

The mind makes models of the universe in order to explain its own perceptions. Mathematics is one of those models.

If we go down right to the root of mathematics — the concept of 'number' — we can see that 'number' is intimately tied to the concept of 'object'. 'Objects' are things that we count and measure using 'numbers'; 'number' has no meaning outside the context of independent and delimited 'objects'. That's why we do not count things like water, air, or electricity: there is no natural object-boundary within these things that sets up individual items to be counted. But philosophically speaking, the concept of 'object' is imposed on the world by the mind according to various characteristics that we perceive to be relevant. It's a good model, but part of the reason it's a good model is that we are fluid in its application. For instance, if we have five diamonds, we expect that number to remain constant — e.g., we would be surprised if we turned around and suddenly discovered there were six diamonds — but if we split one of the diamonds we don't bat an eye about the fact that five suddenly became six. Likewise, if we have five sheep, we expect that one day we will turn around and have more sheep (five becomes seven or eight in the springtime). 'Sheep objects' are numerically fluid, 'diamond objects' are not (except when they are).

I'm fond of Marino Klisovich's answer as well (that mathematics is a language), but mostly through Wittgenstein's understanding of language, which points back at models in the sense I'm talking about.

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