To use Descartes’ famous example, a mental image of a chiliagon (a 1,000-sided figure) cannot be clearly distinguished from a mental image of a 1,002-sided figure, or even from a mental image of a circle. But the concept of a chiliagon is clearly distinct from the concept of a 1,002-sided figure or the concept of a circle. — Wayfarer

Most interesting! — Ms. Marple

Conceptually distinguishable (rationalism) but perceptually not (empiricism).

I wonder about the extent of the conflation among distinct objects that occur due to the low-resolution of our senses?

The prediction of the motion of a drop of water in a rotating spherical mass:

Perception smoothes the grainy world structure. The water feels like a continuous stuff.

Perception smoothes the grainy world structure. The water feels like a continuous stuff. — Landoma1

Like how AI is used to "correct" images on telescopes here on earth and in outer space?

Do we get fooled by AI?

Do we get fooled by AI?

What if we are AI? we just don't know it (yet)! Vide creationism, simulation hypothesis.

Conceptually distinguishable (rationalism) but perceptually not (empiricism). — Agent Smith

Correct. One of your sporadically insightful observations. :wink:

What if we are AI? — Agent Smith

Then we're fucked! Do you really think consciousness can be programmed?

Conceptually distinguishable (rationalism) but perceptually not (empiricism). — Agent Smith

No. Concept and percept are not separable or even two really existing categories. The distiction is purely theoretical.

The main issue is Eugene Wigner's The Unreasonable Effectiveness of Mathematics in the Natural Sciences

As regards the unreasonable effectiveness of mathematics, it is initially truly amazing that from sitting on a train using Richard Hamming's thought experiment, I can disagree with Aristotle and agree with Galileo that heavy objects should fall at the same speed at lighter ones, not only on the train but universally on the far side of the universe.

However, the starting position is the amazing regularity and invariance of what we call "the laws of nature". Given such regularity, the "laws of nature" applicable on the train will also be applicable on the far side of the universe.

The fact that I can codify and quantify using reason, logic and mathematics the "laws of nature" that I observe on the train, and still be applicable on the far side of the universe, is not a measure of success of my reasoning, logic and mathematics, but rather is a measure of the regularity and invariance of the "laws of nature".

Without such regularity and invariance in "the laws of nature", our reasoning, logic and mathematics would count for nothing.

IE, mathematics is only effective because of the unreasonable regularity in the "laws of nature"

An associated issue is the question as to whether universal concepts such as "north of" and "whiteness" are or are not dependent on thought. Are they discovered in the external world or invented in the mind ?

Wayfarer - page 2 - The second, and crucial, point, is that such qualities or relations or whatever they are, (Bertrand Russell's "north of", "whiteness") are not dependent on thought, but they can only be perceived by thought.

RussellA - page 2 - If universals don't exist in a mind-independent world, yet we can perceive them in our minds, then they must have been created in the mind. As universals have been created by thoughts in the mind, they must be dependent for their existence on thoughts in the mind.

IE, not agreed yet.

You're defending the empiricist view that all concepts are derived from experience — Wayfarer

Yes. In the sense that concepts are created in the mind based on observations of the external world, not that concepts ontologically exist in a mind-independent world.

Quite right! That is the point at issue, which here you appear to be conceding. — Wayfarer

I agree that the mind uses the inventive power of intellectual reasoning, and part of its inventive power is in the invention of universal concepts, such as "north of".

The 'law of the excluded middle' didn't come into existence when it was discovered by h. sapiens; it would be true in all possible worlds — Wayfarer

SEP - Disjunction - The law of excluded middle (LEM) states that any proposition of the form (ϕ∨¬ϕ) is logically valid. The semantic principle of bivalence states that every proposition is either true or false (and not both).

SEP - Structured Propositions - For example, when a German speaker utters the sentence ‘Schnee ist weiss’ and an English speaker utters the sentence ‘Snow is white’, they have said the same thing by uttering the sentences they did.
The proposition is taken to be the thing that is in the first instance true or false

IE, as propositions don't exist in a mind-independent world, and as the "law of excluded middle" is based on propositions, the "law of excluded middle" doesn't exist in a mind-independent world.

This is the point of the a priori nature of the pure concepts of reason in Kant — Wayfarer

Exactly, a priori in the mind.

So universal concepts are not created by thought, but can only be discerned by a rational intellect — Wayfarer

If universal concepts were not created by thought, then the universal concepts of love and hate could be discovered in a mind-independent world.

If universal concepts were not created by thought, then the universal concepts of love and hate could be discovered in a mind-independent world. — RussellA

Which can be discovered indeed

My point is that dead matter seems to obey many mathematical structure. If this wasn't the case, chaos would rule supreme. In experiments, reality is arranged to fit the formula.

If universals don't exist in a mind-independent world, yet we can perceive them in our minds, then they must have been created in the mind. — RussellA

I think before going further, you should explain further what you mean by your term 'ontologically exist'.

My point is that dead matter seems to obey many mathematical structure — Landoma1

Mathematics obeys matter, rather than matter obeys mathematics

From observations of the world, we arrive at the belief that there is a regularity in what we call "the laws of nature". We are able to justify such a belief through further experimentation.

In order to model what we believe to be the intrinsic regularity discovered within the "laws of nature", we invent mathematical systems also having intrinsic regularity.

If our intrinsically regular mathematical systems prove to be effective in predicting future states of affairs, then we can infer that the "laws of nature" are also intrinsically regular. We can never prove that the "laws of nature" are intrinsically regular, as this leads into Hume's problem with inductive reasoning. We can never know that the "laws of nature" are intrinsically regular, as knowledge requires a justified true belief, and the truth is beyond what we can inductively reason.

The "unreasonable" effectiveness of mathematics is a strong indication that within the "laws of nature" there is an inherent regularity.

The "unreasonable" effectiveness of mathematics is a strong indication that within the "laws of nature" there is an inherent regularity. — RussellA

Which renders math reasonably effective.

I think before going further, you should explain further what you mean by your term 'ontologically exist'. — Wayfarer

The mind is of a different kind to the mind-independent world

Realism is the belief that the world comprises the mind and a mind-independent world.

Although the mind is part of the world, my belief is that the the nature of the mind is different to the nature of the mind-independent world. FH Bradley's Regress Argument persuades me that relations don't ontologically exist in the mind-independent world, whilst the Binding Problem and Kant's Unity of Perception do persuade me that relations do ontologically exist in the mind.

Although everything in the mind-independent world exists in the mind, such as matter and the forces between them, there are some things that exist in the mind but not the mind-independent world, such as concepts, unicorns, apples, numbers, universals, abstracts, love and hate, ethics, pain and pleasure, fictional characters, ghosts, gods, relations, etc.

Therefore, there is a set of things that ontologically exist in the mind, and a different set of things that ontologically exist in the mind-independent world, though the sets do overlap.

IE, in the mind-independent world, quarks and the weak nuclear force do ontologically exist, but love and hate don't.

:smile:

Then we're fucked! Do you really think consciousness can be programmed? — Landoma1

I hope not! Consciousness has, to my reckoning, many facets to it; the logic has been replicated (on computers) but not duplicated, if you catch my drift.

No. Concept and percept are not separable or even two really existing categories. The distiction is purely theoretical. — Landoma1

:chin: :snicker:

Correct. One of your sporadically insightful observations. :wink: — Wayfarer

:lol: I wouldn't want to cause an outbreak!

The mind is of a different kind to the mind-independent world

Realism is the belief that the world comprises the mind and a mind-independent world. — RussellA

I question the coherence of the idea of a 'mind-independent world', but I don't think I'll pursue it. (I have some familiarity with F. H. Bradley whom as I understand it was one of the last of the British Idealists, in other words, he did not subscribe to the doctrine that reality comprises a plurality of really existing mind-independent objects, as can be seen here.)

It's a mystery to me. All I know is we mathematicians observe physical phenomena and extract and abstract patterns.

It's a mystery to me. All I know is we mathematicians observe physical phenomena and extract and abstract patterns. — jgill

Thank you for your service.

I think all participants here know about the statement of the unreasonable effectiveness of mathematics. Shouldn't we, rather, speak of it's reasonable effectiveness? I can't see nothing unreasonable about it and can't even imagine how else it could be. — Landoma1

First time reading the essay, myself. A handful of quotes from it:

...Every empirical law has the disquieting quality that one does not know its limitations... We may lose interest in the "ultimate truth,"... Such a situation would put a heavy strain on our faith in our theories and on our belief in the reality of the concepts which we form. It would give us a deep sense of frustration in our search for what I called "the ultimate truth."

The language he uses is deeply religious throughout. These were moments where I thought the religious nature of his appeals were apparent -- it may not be a Christian religion, but the religion of the philosophers -- in the God that thinks itself and brings order to nature, or whatever formulation he may prefer.

Which isn't to say he is wrong, I think. What I would say is that religious appeals are only effective among believers. His is the wonder of a scientist with a passion for something I don't think even exists -- "ultimate truth" or "foundations" as he is also often reasoning within. He's sort of pulling a couple of transcendental moves along the way, really -- without the empirical law of epistemology, no physics is possible. Physics exists, therefore....

But for myself I prefer to focus on the multiplicity of science. And there I think I'd actually wend a path between yourself and Wigner. For where you say:

Let's not forget though where we apply it. To dead Nature. In the human realm it seems unreasonable if effective indeed

I wouldn't put any caveats on the usage of mathematics to understand human beings just because it is "alive" whereas matter is "dead" -- we have a theory of evolution, after all, and we use mathematics in biology, so there's no need to think math can't help in understanding life (not that you said this, but you are asserting that nature is dead, so "life" naturally springs to my mind as the antipode). Humans are just the animal that talks too much and thinks such talking is really special, so we could -- with effort -- come to understand the human animal in more precise terms than that.

And further, I think there is something curious about math. I just don't think it's religious, or tied to ultimate truth, or the scientists' quest for the One Pure Description.

I think "Why does mathematics help human beings?" a reasonable and interesting question that seems to me to be a pretty close approximation to what Wigner is mentally ogling in his essay -- I often wonder about the nature of math in relation to nature, I just don't think placing that question in the realm of the mystical or religious to really get me going. I'm not religious.

The language he uses is deeply religious throughout — Moliere

It's funny you say that - his Wikipedia page says he was a convinced atheist. Maybe the fact that it reads as 'religious' is because the kind of mathematical Platonism he seems to be suggesting goes against the grain of philosophical naturalism. There's a remark in another essay about philosophy of maths that I've read, saying 'Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” (i.e. like numbers) makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.'

Mathematics can't make sense of fluid turbulence! :snicker:

Anyone have an idea why?

Is this a case of the nonmathematical nature of reality or is it just that we aren't smart enough?

The language he uses is deeply religious throughout. — Moliere

There is no indication in the article that Wigner proposes the mystical or religious to explain why our theories work so well.

Wigner wrote - "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve".

Metaphors are commonly used in science, such as: evolution by natural selection, F = ma, the wave theory of light, DNA is the code of life, the genome is the book of life, gravity, dendritic branches, Maxwell's Demon, Schrödinger’s cat, Einstein’s twins, greenhouse gas, the battle against cancer, faith in a hypothesis, the miracle of consciousness, the gift of understanding, the laws of physics, the language of mathematics, deserving an effective mathematics, etc.

IE, it would be more true to say that the language he uses is deeply metaphorical rather than religious.

It's funny you say that - his Wikipedia page says he was a convinced atheist. Maybe the fact that it reads as 'religious' is because the kind of mathematical Platonism he seems to be suggesting goes against the grain of philosophical naturalism. There's a remark in another essay about philosophy of maths that I've read, saying 'Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” (i.e. like numbers) makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.' — Wayfarer

I'd say that the reason it reads religiously is that it reminds me of the God of the philosophers -- so he may think he was an atheist, but in the essay he admits that these are articles of faith. "Religious", after all, is much wider than a/theism.

Wigner wrote - "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve" — RussellA

it would be more true to say that the language he uses is deeply metaphorical rather than religious. — RussellA

If you prefer metaphorical, then that's fine. My point remains -- the metaphor works for believers in ultimate truth.

My point remains -- the metaphor works for believers in ultimate truth. — Moliere

I agree that as a metaphor, the "ultimate truth" works for atheists in their belief of an "ultimate truth".

The language he [Wigner] uses is deeply religious throughout — Moliere

It's funny you say that - his Wikipedia page says he was a convinced atheist — Wayfarer

Interesting observations. My ex-wife's father, a friend of Wigners, was an architect and intellectual in Hungarian society. He had no use for organized religion, but was something of a disciple of Tielhard de Chardin, an intellectual and Catholic priest who advanced the idea of an Omega Point, toward which the world moves and reaches in its final days. A curious blend of science and something like religion.

. He had no use for organized religion, but was something of a disciple of Tielhard de Chardin, an intellectual and Catholic priest who advanced the idea of an Omega Point, toward which the world moves and reaches in its final days. — jgill

that is indeed interesting. I think overall that what has happened is that religion has 'burst its banks', i.e. overflowed the boundaries that had been set up for it by the Church. All of the Biblical symbolism of tares and wheat and flocks and blood sacrifice which are natural to an early agrarian culture make no sense in the post-industrial landscape, but there's a deeper level of meaning which flows on regardless.

Kurt Godel apparently developed a rationalist 'proof of God' argument towards the end of his life (ref). He too was religiously unaffiliated, but also a mathematical Platonist, as many physicists are.

Ever since I began to think about it, I've held that numbers and basic geometrical principles and the like are real, in that they're the same for anyone who can grasp them. So they're not dependent on your or my mind, but can only be grasped by a rational mind. Secondly, that because reason uses these to interpret and organise experience, then they are fundamental elements of lived reality, not in the way that objects and energy are, but as fundamental constituents of the human 'life-world'.