It was Eugene Wigner who spoke of the "unreasonable effectiveness of math". Nature has dead and alive elements. Many deed phenomena (which doesn't mean they don't contain at least the seeds of life) behave in fixed patterns, contrary to living phenomena. For example, the principle of least action applies to dead matter but not to life. — Landoma1

"Dead" matter means it was once alive, yes?

I missed a golden opportunity years ago to exchange letters with Wigner or talk with him. My ex-wife's father, who lived in Montana, was a Hungarian aristocrat who corresponded regularly with Wigner. I could have discussed this very issue, being an academic math person. Sometimes we skirt by greatness without recognizing an opportunity.

From Wigner's paper:

The principal emphasis is on the invention of concepts

, and he goes on to observe that we would run out of interesting theorems were it not for the creation or discovery of new concepts.

What would you had asked Him? To give back your wife? Just kidding! : :joke:

Good question. I was much younger and I think an associate professor at the time, and my interests then might not have been my interests now - hard to recall. But probably I would have discussed the origins of mathematical concepts, those ideas that seem to pop out of nowhere and can become so important. Without such breakthroughs math would stagnate. An example is the notion of metric spaces, an abstraction or generalization of Euclidean distance.

I like the Hundertwasser houses. He seems to have played with standard Euclidean bricks.

Wagner, not wigner

Wagner, not wigner — jgill

:smile:

Mozart has some tasty chocolate balls!

I still don't see why that's unreasonable. — Landoma1

If you don't see the point of Wigner's essay, there's not a lot of purpose me trying to explain it again.

Referring to Eugene Wigner's - The Unreasonable Effectiveness of Mathematics in the Natural Sciences

Mathematical systems are invented and then discarded if discovered to be ineffective

(I hope @jgill agrees)

As Wigner wrote: "I would say that mathematics is the science of skilful operations with concepts and rules invented just for this purpose. The principal emphasis is on the invention of concepts".

I can invent the elementary mathematical system such that 1 + 1 = 3. Many mathematical systems can be invented. The mathematical system whereby 1 + 1 = 2 has also been invented.

As it has been discovered through observation of the external world that the mathematical system 1 + 1 = 2 is more useful than the mathematical system 1 + 1 = 3, the former has been kept and the latter discarded.

IE, Wigner was correct when he wrote about "the unreasonable effectiveness of mathematics in the natural sciences", because any mathematics that has been shown not to be unreasonably effective has been discarded

Reason, logic and mathematics require regularities in the external world

Wigner wrote - The laws of invariance of physical theories.........regularities in the events in the world around us which can be formulated in terms of mathematical concepts with an uncanny accuracy"

If there were not inherent regularities in the world, we could not predict that two rocks dropped at the same time from the same height reach the ground at the same time, regardless on whether on the Earth or the far side of the Universe.

"Laws of nature" exist only in the mind and not the external world

Wigner wrote "It is not at all natural that "laws of nature" exist, much less that man is able to discover them"

As concepts such a tables, apples, governments, ethics, laws of nature, etc require relationships between their parts, and relations have no ontological existence in the external world (see FH Bradley) but only in the mind (see the Binding Problem and Kant's Apperception and its Unity), such concepts exist only in the mind and not the external world.

Our beliefs are always "unreasonably effective"

Wigner wrote: We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became sceptical concerning the uniqueness of the coordination between keys and doors".

In general, our beliefs in tables, apples, governments, ethics, laws of nature, mathematics, etc are always "unreasonably effective" and well suited to our understanding because beliefs are self-referential.

On the one hand, I have a concept of or a belief about a law of nature existing in the external world, and on the other hand, a law of nature is a concept or belief that only exists in the mind and not the external world.

IE, even if our belief is wrong, we still believe it.

I consider the sentence "One plus one equals two" and now I pretend to be astonished that when I count actual things I find that one of them plus another one of them equals two of them. Then I imagine that this is a suspicious and unreasonably co-incidental fit of arithmetic to the world and of the world to arithmetic. But I do all this to what end - or is it just a pastime? It does not seem to show anything. Perhaps we invented arithmetic because we find it a useful tool. Perhaps we discovered arithmetic because it's there to be discovered. Feigning astonishment does not seem to add anything.

As concepts such a tables, apples, governments, ethics, laws of nature, etc require relationships between their parts, and relations have no ontological existence in the external world (see FH Bradley) but only in the mind (see the Binding Problem and Kant's Apperception and its Unity), such concepts exist only in the mind and not the external world. — RussellA

Amazing that inventions work so well, then. Or maybe they're all in the mind as well?

Amazing that inventions work so well, then. Or maybe they're all in the mind as well? — Wayfarer

What do you make of the phenomenologist's position, articulated so well by Joshs?

But the fact that there is structure to the world does not mean that the world comes to our awareness packaged an ‘inherent’ way that is already mathematical. Nature became mathematizable when we contributed our own peculiar interpretive structures to it.
As you can see, I’m a mathematical constructivist, not a platonist. — Joshs

In philosophy there seems to be a constant game of 'did we discover it or invent it?' - whether we are talking math or morality. I know how you feel about Platonism, but what do you make of the view from phenomenology?

I do want to know a bit more about Husserl's philosophy of maths, although I fear that it's a very dense subject. But I don't feel comfortable with the idea that number is a kind of mental projection. I'm also mindful of the way that mathematical physics is so powerfully predictive. It's not a game or a scheme.

Amazing that inventions work so well, then. Or maybe they're all in the mind as well? — Wayfarer

Take the wheel as an example. As with all inventions they were invented in the mind, and then made into a physical thing of perhaps wood and steel, a circular rim supported by spokes revolving around an axle.

The parts of the wheel physically exist in the world, and once created, exist independently of whether anyone is thinking of them or not. This is compatible with Realism, in that there is a mind-independent world of matter and forces.

For the Naive Realist, our sense provide us with a direct awareness of the wheel as it really is.

For the Indirect Realist, we do not perceive the external world as it really is, but are only aware of a representation of the external world. Our awareness is of the concept of the wheel, not the "wheel" itself.

The concept of a wheel definitely exists in the mind, otherwise we would not be having this conversation. The question is, do wheels ontologically exist in the external world ? A wheel is a spatial relation between its rim, spokes and axle. Whether wheels ontologically exist in the external world depends on whether relations ontologically exist in the external world.

I am persuaded by FH Bradley's Regress Argument that relations don't ontologically exist in the external world, meaning that wheels don't ontologically exist in the external world.

If one is going to argue that wheels do ontologically exist in the external world, then this will require a justification that relations also ontologically exist in the external world.

Is there any reasoned argument that relations do ontologically exist in the external world ?

Is there any reasoned argument that relations do ontologically exist in the external world ? — RussellA

https://thephilosophyforum.com/discussion/comment/650566

You wrote:
We have to show that relations exist.
Glasgow is west of Edinburgh - so we are told.
It could mean that if we turn over the whole universe item by item we will find at least one thing that is to the west of another thing.
There is a case - at least one case - of something being to the west of something else.
From which if finally follows that relations exist...........we can see that relations have ontological existence.

As regards the ontological existence of relations, there are two possibilities - either i) relations don't exist and after turning over the whole universe item by item we won't find at least one thing that is to the west of another - or ii) relations do exist and after turning over the whole universe item by item we will find at least one thing that is to the west of another

However, the problem with your statement "There is a case - at least one case - of something being to the west of something else" is that the statement can only be made on the assumption that relations do exist.

It is equivalent to saying that if relations exist then relations exist.

Nature is not dead. — Jackson

It is.

If you don't see the point of Wigner's essay, there's not a lot of purpose me trying to explain it again. — Wayfarer

It's not I don't understand his irony. I think it's unreasonable. Why should it be unreasonable that math predicts anti particles? It's a logical, so reasonable, consequence of the Dirac equation.

But you’re assuming just the thing that is in question. You know, ‘the task of philosophers is to wonder at what men think ordinary’. Philosophy asks questions about many things that you ordinarily take for granted. That mathematics can predict things that not only are not known, but of a kind of which was never even previously imagined, like anti-matter. So a philosophical response to that is not, I suggest, ‘so what?’

As with all inventions they were invented in the mind, and then made into a physical thing of perhaps wood and steel, a circular rim supported by spokes revolving around an axle. — RussellA

But you don't know that. It's quite feasible that the wheel was invented because some Cro-Magnon discovered that you could roll a big rock on logs. Basic empiricism.

I am persuaded by FH Bradley's Regress Argument that relations don't ontologically exist in the external world, meaning that wheels don't ontologically exist in the external world. — RussellA

I'm not familiar with Bradley's Regress Argument, nor with the meaning of 'ontologically exist'. But at least you're prepared to discuss universals, which few here are, so I will persist.

It could mean that if we turn over the whole universe item by item we will find at least one thing that is to the west of another thing.
There is a case - at least one case - of something being to the west of something else.
From which if finally follows that relations exist...........we can see that relations have ontological existence. — RussellA

I think that you're saying that statements about what exists can only be validated against instances that might be encountered in the sensable universe. That for something to exist, it has to be 'out there somewhere' - you have to find an instance of it in order for it to 'ontologically exist'. I'm guessing that's what you mean by that rather awkward phrase.

Bertrand Russell provides an example in Problems of Philosophy, which I'm guessing you're familiar with, given the particular example that you've cited involving Glasgow. The way he puts it is like this (and please forgive the lengthy quotation):

Consider such a proposition as 'Edinburgh is north of London'. Here we have a relation between two places, and it seems plain that the relation subsists independently of our knowledge of it. When we come to know that Edinburgh is north of London, we come to know something which has to do only with Edinburgh and London: we do not cause the truth of the proposition by coming to know it, on the contrary we merely apprehend a fact which was there before we knew it. The part of the earth's surface where Edinburgh stands would be north of the part where London stands, even if there were no human being to know about north and south, and even if there were no minds at all in the universe. ...We may therefore now assume it to be true that nothing mental is presupposed in the fact that Edinburgh is north of London. But this fact involves the relation 'north of', which is a universal; and it would be impossible for the whole fact to involve nothing mental if the relation 'north of', which is a constituent part of the fact, did involve anything mental. Hence we must admit that the relation, like the terms it relates, is not dependent upon thought, but belongs to the independent world which thought apprehends but does not create.

This conclusion, however, is met by the difficulty that the relation 'north of' does not seem to exist in the same sense in which Edinburgh and London exist. If we ask 'Where and when does this relation exist?' the answer must be 'Nowhere and nowhen'. There is no place or time where we can find the relation 'north of'. It does not exist in Edinburgh any more than in London, for it relates the two and is neutral as between them. Nor can we say that it exists at any particular time. Now everything that can be apprehended by the senses or by introspection exists at some particular time. Hence the relation 'north of' is radically different from such things. It is neither in space nor in time, neither material nor mental; yet it is something. ...

It is largely the very peculiar kind of being that belongs to universals which has led many people to suppose that they are really mental. We can think of a universal, and our thinking then exists in a perfectly ordinary sense, like any other mental act. Suppose, for example, that we are thinking of whiteness. Then in one sense it may be said that whiteness is 'in our mind'. ... In the strict sense, it is not "whiteness" that is in our mind, but "the act of thinking of whiteness". The connected ambiguity in the word 'idea', which we noted at the same time, also causes confusion here. In one sense of this word, namely the sense in which it denotes the object of an act of thought, whiteness is an 'idea'. Hence, if the ambiguity is not guarded against, we may come to think that whiteness is an 'idea' in the other sense, i.e. an act of thought; and thus we come to think that whiteness is mental. But in so thinking, we rob it of its essential quality of universality. One man's act of thought is necessarily a different thing from another man's; one man's act of thought at one time is necessarily a different thing from the same man's act of thought at another time. Hence, if whiteness were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts.

We shall find it convenient only to speak of things existing when they are in time, that is to say, when we can point to some time at which they exist (not excluding the possibility of their existing at all times). Thus thoughts and feelings, minds and physical objects exist. But universals do not exist in this sense; we shall say that they subsist or have being, where 'being' is opposed to 'existence' as being timeless. — Bertrand Russell, The World of Universals

There are very many crucial points made in this passage, of which I'll point to a few. One is the distinction between the existence of universals and of sensable objects. As Russell says, neither 'north of' nor 'whiteness' exist - rather they subsist or 'have being' - rather an awkward expression, but it's rather a difficult point. The second, and crucial, point, is that such qualities or relations or whatever they are, are not dependent on thought, but they can only be perceived by thought. Both of which point to what I regard as an absence or lack in the current philosophical lexicon, with respect to the distinction between 'being' and 'existence'. It's the tip of a very large iceberg.

It's not "so what". That's how you interpret it. It's "that's how it is". If matter behaves regularly it comes as no surprise that math describes it well. Math is about quantitative regularities and interdependences. The Dirac equation is an example. If you work with the equation you work with a naturally occurring regularity. So you bump into the positron.

That's how you interpret it. — Landoma1

No, that's what you've said. You say in your OP, you can't imagine it being any different. That's because you live in a society that has made these discoveries. If you lived a hundred years ago you couldn't even imagine such a discovery. But, yeah, so what.

If you lived a hundred years ago you couldn't even imagine such a discovery — Wayfarer

No, of course not. That's stating the obvious. But a 100 years ago I could have used general relativity to predict unobserved phenomena. That's how it works. You make predictions with math. Like the Higgs boson was inferred 60 years ago. Like many other phenomena are calculated before discovery. How else could it function? If this wasn't the case, physics would have abandoned .math long ago. The eight-fold way.

But you don't know that. It's quite feasible that the wheel was invented because some Cro-Magnon discovered that you could roll a big rock on logs. Basic empiricism. — Wayfarer

Concepts are invented not discovered

One day, a Cro-Magnon on a walk through the forest happened to observe a big flat rock that had fallen from the side of a mountain rolling along on some logs on the ground. Let us call the Cro-Magnon George.

George had made a particular empirical observation and discovered something in the world, in that a flat rock can continue to move freely along the ground as long as it is supported by round logs.

George was able to make the intellectual leap from the particular to the general, and reason that any flat object may move more freely along the ground as long as it is supported by round objects. George using reason was able to use a particular observation to arrive at a universal concept, the concept of the wheel.

If concepts existed in a mind-independent world then they would be "out there somewhere"
and discoverable. However, if that were the case, two people independently observing an object, for example a rock, should be able to write down all concepts discoverable within the object, in which case, when compared, their lists should be the same. Concepts in objects cannot be discovered by observation alone, but require the inventive power of reasoning using the intellect.

IE, because the concept of the wheel does not exist in either a mind-independent rock, log or relation between the two, George could not have discovered the concept from an empirical observation. But as the concept of the wheel exists in George's mind, and because it cannot have been discovered from an empirical observation, it must have originated in George's mind, been invented in the mind of George.

This conclusion, however, is met by the difficulty that the relation 'north of' does not seem to exist in the same sense in which Edinburgh and London exist. — Bertrand Russell, The World of Universals

"North of" and London exist in exactly the same way

The object Edinburghlondon has the related parts i) Edinburgh and ii) London
London has the related parts i) north of the Thames and ii) south of the Thames
South of the Thames has the related parts i) built up areas and ii) trees and grasslands
Built up areas have the related parts i) buildings and ii) roads
Roads have the related parts i) paving and ii) junctions
Paving has the related parts - etc

Russell writes that " the relation 'north of' is radically different from such things" as London, in that "north of" subsists whilst London exists. Yet London only exists as a relation between its parts north of the Thames and south of the Thames.

IE, the object London depends on its existence on relations between parts - the area north of the Thames and the area south of the Thames. If, according to Russell, relations don't exist in a mind-independent world, then it follows from Russell's own argument that neither can London exist as a mind-independent object.

That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts. — Bertrand Russell, The World of Universals

Universals are thoughts

A group of people may agree to share a common language. It may be agreed within the group that white objects are linked with the word "white". In the world being observed by the group, not only do white objects physically exist, but also and the word "white" physically exists as an object.

Each individual may develop their private concept of "whiteness" by observing white objects in the world. But also each individual may also link their private concept of "whiteness" with the associated public word "whiteness"

It is true as Russell writes "One man's act of thought is necessarily a different thing from another man's" - in that my concept of "whiteness" is of necessity not identical to yours.

But it is not true as Russell wrote that "if whiteness were the thought as opposed to its object, no two different men could think of it", as one person's private concept of "whiteness" is linked to everyone else's private concept of "whiteness" through the public linguistic object "whiteness".

IE, universals, such as the concept of "whiteness" not only exist in the mind as a private thought, but also are universally understood by a group sharing a common public language.

One is the distinction between the existence of universals and of sensable objects. — Wayfarer

Universals and sensables are of the same kind

There are sensable objects such as Edinburgh and London and universals such as "north of" and "whiteness".

IE, a sensable object such as London is no more that a relation between its parts, and if relations are universals, then sensable objects cannot be argued to be of a different kind to the universals from which they are comprised.

such qualities or relations or whatever they are, are not dependent on thought, but they can only be perceived by thought — Wayfarer

Universals are dependant on thought

It has been well-argued that universals such as whiteness, apples, houses, governments can not be explained as Platonic Forms that exist in a mind-independent world.

Universals can be perceived by thought, in that we can discuss them.

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, universals are dependent on thought and can be perceived by thought.

It is equivalent to saying that if relations exist then relations exist. — RussellA

There is one step in between. It's this:

If relations do not exist, then it is not the case that something is to the west of something else.
Glasgow is to the west of Edinburgh.
Therefore (at least one) relation exists.

The first premiss does not assume that relations exist. It suggests a criterion for finding out whether they exist or not. The suggested criterion is to go and look at the world to see whether anything is to the west of anything else. I cannot think of a better criterion, but perhaps there is. So it's a challenge - if that's not what it is for a relation to exist, what should the criterion be?

If relations do not exist, then it is not the case that something is to the west of something else.
Glasgow is to the west of Edinburgh.
Therefore (at least one) relation exists. — Cuthbert

This leads to:

Statement 1 = If relations do not exist, then it is not the case that Glasgow is to the west of Edinburgh

Statement 2 = Relations exist, and It is the case that Glasgow is to the west of Edinburgh.

The conclusion that "therefore (at least one) relation exists" of necessity follows from statement 2 rather than prove anything about the ontological existence or not of relations.

Confirmed predictions are not a guarantee that the model you uses is right. Epicycle theory (or Fourier analysis, for that matter), was based on imaginaries, though predictions were fairly good.

The OP's question is marvelous!

Why is math's utility "unreasonable"? I guess the answer has to do with scope (cross-domain applicability) and strength (the certainty that it guarantees).

What should worry us:

1. Maslow's hammer

2. The Streetlight Effect

A group of people may agree to share a common language. It may be agreed within the group that white objects are linked with the word "white". In the world being observed by the group, not only do white objects physically exist, but also and the word "white" physically exists as an object. — RussellA

You're defending the empiricist view that all concepts are derived from experience. (Some of the following is cribbed from Edward Feser)

Your initial example of a wheel was poorly chosen, the universal concept that is nearest is actually the circle. A circle is 'a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the centre)' as a matter of definition. As such, that is a concept which is discoverable by any mind capable of understanding such concepts. Something similar can be said of other geometrical primitives (squares, triangles and so on.) And these are concepts, not mental images - any mental image you can form of a triangle will be an image of an isosceles , scalene, or equilateral triangle, of a black, blue, or green triangle, etc, whereas the abstract concept "triangularity" applies to all triangles without exception. The concepts that are the objects of intellectual activity are universal, while mental images and sensations are essentially particular. Any mental image you can form of a man is always going to be of a particular man - tall, short, fat, thin, blonde, redheaded, bald, or whatever. But the concept "man" applies to every man.

Second, mental images are always to some extent vague or indeterminate, while concepts are at least often precise and determinate. 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.

Third, we have many concepts that are so abstract that they do not have even the loose sort of connection with mental imagery that concepts like man, triangle, and crowd have. You cannot visualize triangularity or humanness per se, but you can at least visualize a particular triangle or a particular human being. But we also have concepts -- such as the concepts law, square root, logical consistency, collapse of the wave function, and innumerably many others -- that can strictly be associated with no mental image at all. You might form a visual or auditory image of the English word “law” when you think about law, but the concept "law" obviously has no essential connection whatsoever with that word, since ancient Greeks, Chinese, and Indians had the concept without using that specific word to name it.

If concepts existed in a mind-independent world then they would be "out there somewhere" and discoverable. However, if that were the case, two people independently observing an object, for example a rock, should be able to write down all concepts discoverable within the object, in which case, when compared, their lists should be the same. Concepts in objects cannot be discovered by observation alone, but require the inventive power of reasoning using the intellect. — RussellA

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

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. — RussellA

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, and would remain so, even if h. sapiens were to become extinct. This is the point of the a priori nature of the pure concepts of reason in Kant. So universal concepts are not created by thought, but can only be discerned by a rational intellect. And that is quite in keeping with the mainstream of Western philosophical thought, even if not with today's empiricism - so much the worse for it!

I read this:

"A couple of years later, I read a theoretical paper that predicted the effect that I had seen in my data. It wasn’t a monumental thing at all - just an unexpected result buried in the mathematical description. I scoured my old research notebooks, saw the effect in those preliminary data sets - but only in those incomplete experiments that were essentially feasibility studies for my real project. There was nothing complete enough to publish"

An unexpected result buried in the mathematical description.