Confusing the intuitive with the abstract, the map with the territory, or the signified with the signifier, seems to be a perennial error in human thinking. Even brilliant philosophers and mathematicians get caught up in this error.

For a view on how this error affects modern physics, I recommend this book:

I think the error might stem from the fact that the human mind is actively working in perception, empirical cognition and in abstract thinking. Those three activities found a common source in the human intellect, and so we mistake abstract entities, such as numbers, with real entities such as matter.

Also, math and logic are artificial languages, but within their axioms they are, nonetheless, apodictic and, in a sense, 'beautiful'. They're the most exact branches of human knowledge. We may therefore, unconsciously, lean toward them even when the subject is entirely empirical. Logic and maths are psychologically satisfying disciplines, that is why is so 'natural' to think of everything intuitive as a 'number', as Pythagoras so famously put it.

I'd just note that the Illuminists not only mistake abstract entities for real ones, they invert the two along the lines of metaphysical idealism. They don't add numbers to material reality, but discount both matter and the sciences that study matter, while elevating math and logic.

Personally, I always found math as the opposite of "psychologically satisfying." I can see why having exact answers that work might be satisfying to many, but lots of math (not geometry or arithmetic) felt arbitrary to me when I learned it in school, because there was zero philosophical underpinning given, but only a building up of more and more stipulations.

Many years later, when I came across Smolin's account of math as evocative and game-like, the stipulations made sense. I could understand math not as platonic and Utopian but as pragmatic model-building, comparable to the building up of fantasy worlds, some of which prove immensely useful since their rigor is tied to our experience of natural patterns.

"I could understand math not as platonic and Utopian but as pragmatic model-building, comparable to the building up of fantasy worlds"

Exactly. I think this is why math is so appealing to many philosophers, mathematicians and scientists. So much so, that they can forget that it is a model, a fabrication, despite its multiples applications. Something akin to music. Plato and Leibniz were enraptured by it.

But, in the end, as you say, it is just a "building up of more and more stipulations" with nothing intuitive i.e. empirical. Maybe for this reason it can have the opposite effect on some people, I guess. I didn't particularly like maths in school either.

Didn't Goethe loathe mathematics for precisely this reason?

I think the appeal of mathematicism is that numbers have so far been proven to be irreducible. We can model almost anything mathematically, but all attempts to model math in some alternative language have failed. If you accept the premise of monism, that all of reality is ultimately made up of one thing, then it's easy to conclude that this thing is math. The alternative to monism is dualism, but then you would need to explain how two fundamentally unlike substances could interact with each other

Concerning natural languages like English & Chinese: it should be noted that all natural languages possesses a mathematical structure. I first learned of this when studying old fashioned crypto-analysis, but apparently linguists have known it for a long time. So even if much of reality can be described in English, that very description could be described through mathematics. Mathematics could itself be described in English (as you pointed out), but that description would be limited & garbled because the potential of each language to describe the other is asymmetrical.

You make an interesting point about there being no sufficient reason for the principle of sufficient reason. This is analogous to positivism being nonsense by its own definition & the failure of Aristotle to justify his own logic - a logic that ultimately proved inadequate to describe reality on the quantum level. I guess the problem is that we need to start from somewhere. Every system of thought has to rest upon certain unfalsifiable claims that cannot themselves be justified within that system - Kurt GĂ¶del rigorously demonstrated why this is true of math, but I suspect it's equally true of rationalism, empiricism, ethics & religion. It ultimately comes down to faith. But are all objects of faith equal? Is there a criteria for choosing which axioms to build upon that is not itself baseless & arbitrary?

I’m not actually clear on this claim of yours that math is the deepest language because even though its statements can be translated into natural language, there’s loss of content in those translations. Surely the subjective aspects of natural language are equally lost in any mathematical model of English, French, or Chinese.

All models simplify. In fact, I’m not even sure what a purely mathematical model would be apart from a blueprint, say, of how some syntax works. What else could be modeled in purely quantitative terms, the behaviour of language users? However far math might stretch in those respects, wouldn’t math presuppose the syntax/semantics distinction? How, then, would math be deeper than natural language? Why not say quantitative and qualitative models differ, each having their strengths and weaknesses, and leave it at that, without claiming one is metaphysically deeper than the other?

The link you supplied is about computational models of natural language, but those aren’t equivalent to math, since they draw on “linguistics, computer science, artificial intelligence, math, logic, philosophy, cognitive science, cognitive psychology, psycholinguistics, anthropology and neuroscience, among others.”

I don’t believe I say exactly that rationalism refutes itself in the positivist manner, by being unable to justify itself in its own terms. I say these extreme rationalists are guilty of a performative contradiction, because the principle of sufficient reason is aligned with certain values that aren’t mathematical. Yes, Illuminism isn’t self-justifying in that sense, but I wanted to stress the irony that it’s not just a question of incompleteness, but of the mathematical worldview’s having to bring in dreaded values, ideals, and other subjective elements in particular.

It's true that Leibniz did invoke non-mathematical, non-rational values in his explanation of PSR but though the Illuminists take much inspiration from Leibniz, they reject his theism (or more correctly, they believe his theism was a sham to deceive the Church so he could work in peace). Anyways, I don't think theism is necessary to justify PSR. Personally, I use it on the pragmatic grounds that when I look for sufficient reasons, I tend to find them. Illuminist rhetoric is of course irrational, but pragmatic given how much more effective it is than purely rational discourse. Why a purely rational universe should spew up a species as irrational as humans who reject rational argument in favor of rhetoric is something you must ask them.

I think a promising clue to the question of how quanta can give rise to qualia lies in music. The fact that a certain mathematical pattern of sine waves traveling through the atmosphere can be used to communicate even the most complex of human emotions would be inexplicable if numbers lacked qualia & in fact Pythagoras did assign qualities to numbers.

But I think what you might be getting at is that even if reality were mathematical, this wouldn't explain our own internal experience of it, which is anything but mathematical. I'll admit this is a conundrum for which I have no answer. Any system other than idealism runs up against the rock of consciousness. A brain may be able to encode millions of terrabytes of information just as we encode the same onto computer chips or, before the computer, onto the pages of a book. But a book is just scratches of ink on paper until someone opens it & begins to read. Knowledge presupposes a knower. A calculator, though it can perform lightning fast calculations, has no actual knowledge of numbers; it doesn't know what it's doing because it isn't conscious. Mathematician Roger Penrose studies this problem in The Emperor's New Mind & concludes that there is no way a computer algorithm, no matter how sophisticated, could ever endow an AI with consciousness which would seem to refute the Illuminist's Mathematical Idealism. Penrose then goes on to explain consciousness with the collapse of wave functions, but he looses me there. I understand that some physicists believe that observation (consciousness) collapses wave functions, but I don't see how the reverse could be true.

Other than idealism, I think the only viable option is dualism where consciousness is an independent substance. But unless we can explain how consciousness interacts with non-consciousness, dualism is a dead end.

Mathematics is part of human language. It's arbitrary separate, divide or sum individual objects and then call "reality itself". These things exist variably randomly in nature. If there are x number of apples in given three is doesnt mean the specific number of apples on it is the manifestation of reality itself because it need a observer to do it based on its own subjectivity. Reality is intrinsically logical, why people confuse with mathematics. "My" concept of logics is anything which obey laws and or is systematically coherent. Physico-chemical reality we live is coherent because it is govermned, replicative and constituted by general or basal laws or patterns or levels of significant stereotypical predicability.

Confusing the intuitive with the abstract, the map with the territory, or the signified with the signifier, seems to be a perennial error in human thinking. Even brilliant philosophers and mathematicians get caught up in this error.

ReplyDeleteFor a view on how this error affects modern physics, I recommend this book:

https://www.goodreads.com/book/show/36341728-lost-in-math

I think the error might stem from the fact that the human mind is actively working in perception, empirical cognition and in abstract thinking. Those three activities found a common source in the human intellect, and so we mistake abstract entities, such as numbers, with real entities such as matter.

Also, math and logic are artificial languages, but within their axioms they are, nonetheless, apodictic and, in a sense, 'beautiful'. They're the most exact branches of human knowledge. We may therefore, unconsciously, lean toward them even when the subject is entirely empirical. Logic and maths are psychologically satisfying disciplines, that is why is so 'natural' to think of everything intuitive as a 'number', as Pythagoras so famously put it.

I'd just note that the Illuminists not only mistake abstract entities for real ones, they invert the two along the lines of metaphysical idealism. They don't add numbers to material reality, but discount both matter and the sciences that study matter, while elevating math and logic.

DeletePersonally, I always found math as the opposite of "psychologically satisfying." I can see why having exact answers that work might be satisfying to many, but lots of math (not geometry or arithmetic) felt arbitrary to me when I learned it in school, because there was zero philosophical underpinning given, but only a building up of more and more stipulations.

Many years later, when I came across Smolin's account of math as evocative and game-like, the stipulations made sense. I could understand math not as platonic and Utopian but as pragmatic model-building, comparable to the building up of fantasy worlds, some of which prove immensely useful since their rigor is tied to our experience of natural patterns.

"I could understand math not as platonic and Utopian but as pragmatic model-building, comparable to the building up of fantasy worlds"

DeleteExactly. I think this is why math is so appealing to many philosophers, mathematicians and scientists. So much so, that they can forget that it is a model, a fabrication, despite its multiples applications. Something akin to music. Plato and Leibniz were enraptured by it.

But, in the end, as you say, it is just a "building up of more and more stipulations" with nothing intuitive i.e. empirical. Maybe for this reason it can have the opposite effect on some people, I guess. I didn't particularly like maths in school either.

Didn't Goethe loathe mathematics for precisely this reason?

I think the appeal of mathematicism is that numbers have so far been proven to be irreducible. We can model almost anything mathematically, but all attempts to model math in some alternative language have failed. If you accept the premise of monism, that all of reality is ultimately made up of one thing, then it's easy to conclude that this thing is math. The alternative to monism is dualism, but then you would need to explain how two fundamentally unlike substances could interact with each other

ReplyDeleteConcerning natural languages like English & Chinese: it should be noted that all natural languages possesses a mathematical structure. I first learned of this when studying old fashioned crypto-analysis, but apparently linguists have known it for a long time. So even if much of reality can be described in English, that very description could be described through mathematics. Mathematics could itself be described in English (as you pointed out), but that description would be limited & garbled because the potential of each language to describe the other is asymmetrical.

You make an interesting point about there being no sufficient reason for the principle of sufficient reason. This is analogous to positivism being nonsense by its own definition & the failure of Aristotle to justify his own logic - a logic that ultimately proved inadequate to describe reality on the quantum level. I guess the problem is that we need to start from somewhere. Every system of thought has to rest upon certain unfalsifiable claims that cannot themselves be justified within that system - Kurt GĂ¶del rigorously demonstrated why this is true of math, but I suspect it's equally true of rationalism, empiricism, ethics & religion. It ultimately comes down to faith. But are all objects of faith equal? Is there a criteria for choosing which axioms to build upon that is not itself baseless & arbitrary?

I’m not actually clear on this claim of yours that math is the deepest language because even though its statements can be translated into natural language, there’s loss of content in those translations. Surely the subjective aspects of natural language are equally lost in any mathematical model of English, French, or Chinese.

DeleteAll models simplify. In fact, I’m not even sure what a purely mathematical model would be apart from a blueprint, say, of how some syntax works. What else could be modeled in purely quantitative terms, the behaviour of language users? However far math might stretch in those respects, wouldn’t math presuppose the syntax/semantics distinction? How, then, would math be deeper than natural language? Why not say quantitative and qualitative models differ, each having their strengths and weaknesses, and leave it at that, without claiming one is metaphysically deeper than the other?

The link you supplied is about computational models of natural language, but those aren’t equivalent to math, since they draw on “linguistics, computer science, artificial intelligence, math, logic, philosophy, cognitive science, cognitive psychology, psycholinguistics, anthropology and neuroscience, among others.”

I don’t believe I say exactly that rationalism refutes itself in the positivist manner, by being unable to justify itself in its own terms. I say these extreme rationalists are guilty of a performative contradiction, because the principle of sufficient reason is aligned with certain values that aren’t mathematical. Yes, Illuminism isn’t self-justifying in that sense, but I wanted to stress the irony that it’s not just a question of incompleteness, but of the mathematical worldview’s having to bring in dreaded values, ideals, and other subjective elements in particular.

It's true that Leibniz did invoke non-mathematical, non-rational values in his explanation of PSR but though the Illuminists take much inspiration from Leibniz, they reject his theism (or more correctly, they believe his theism was a sham to deceive the Church so he could work in peace). Anyways, I don't think theism is necessary to justify PSR. Personally, I use it on the pragmatic grounds that when I look for sufficient reasons, I tend to find them. Illuminist rhetoric is of course irrational, but pragmatic given how much more effective it is than purely rational discourse. Why a purely rational universe should spew up a species as irrational as humans who reject rational argument in favor of rhetoric is something you must ask them.

DeleteI think a promising clue to the question of how quanta can give rise to qualia lies in music. The fact that a certain mathematical pattern of sine waves traveling through the atmosphere can be used to communicate even the most complex of human emotions would be inexplicable if numbers lacked qualia & in fact Pythagoras did assign qualities to numbers.

But I think what you might be getting at is that even if reality were mathematical, this wouldn't explain our own internal experience of it, which is anything but mathematical. I'll admit this is a conundrum for which I have no answer. Any system other than idealism runs up against the rock of consciousness. A brain may be able to encode millions of terrabytes of information just as we encode the same onto computer chips or, before the computer, onto the pages of a book. But a book is just scratches of ink on paper until someone opens it & begins to read. Knowledge presupposes a knower. A calculator, though it can perform lightning fast calculations, has no actual knowledge of numbers; it doesn't know what it's doing because it isn't conscious. Mathematician Roger Penrose studies this problem in

The Emperor's New Mind& concludes that there is no way a computer algorithm, no matter how sophisticated, could ever endow an AI with consciousness which would seem to refute the Illuminist's Mathematical Idealism. Penrose then goes on to explain consciousness with the collapse of wave functions, but he looses me there. I understand that some physicists believe that observation (consciousness) collapses wave functions, but I don't see how the reverse could be true.Other than idealism, I think the only viable option is dualism where consciousness is an independent substance. But unless we can explain how consciousness interacts with non-consciousness, dualism is a dead end.

Mathematics is part of human language. It's arbitrary separate, divide or sum individual objects and then call "reality itself". These things exist variably randomly in nature. If there are x number of apples in given three is doesnt mean the specific number of apples on it is the manifestation of reality itself because it need a observer to do it based on its own subjectivity. Reality is intrinsically logical, why people confuse with mathematics. "My" concept of logics is anything which obey laws and or is systematically coherent. Physico-chemical reality we live is coherent because it is govermned, replicative and constituted by general or basal laws or patterns or levels of significant stereotypical predicability.

ReplyDelete