Applicability of Maths to the Universe

[Belacqua]

It's Penrose, so I'm going to take him seriously.

He's a smart guy! 

Here's an article I found just now, by coincidence. It touches on a number of issues related to the topic here, I think. It's particularly interesting to me how Newton changed what science was looking for, by giving up on material causal explanations and looking instead for mathematization. So in a way, science says it has understood the material when it stops looking at the material and changes it into an abstraction -- math. 

https://americanaffairsjournal.org/2020/...t-meaning/

Much of what the writer refers to in passing is described in detail in Burtt's classic book The Metaphysical Foundations of Modern Science. 

https://www.amazon.com/Metaphysical-Foun...oks&sr=1-1

Which may be pirated here:

http://gen.lib.rus.ec/book/index.php?md5...7F3D3AD067

As an early critic of the Enlightenment, William Blake accused Newton of valuing the abstracted understanding over the genuine material -- of claiming that truth is in the formula rather than in the thing itself.

[polymath257]

There are two sides for mathematics: math as mathematicians do it, and math as other people use it.

For a mathematician, mathematics is a formal system, defined by axioms and rules of inference. The goal, for a mathematician, is to produce 'beautiful mathematics'. But it is a characteristic of that formal system that it allows for a great many different subsystems. So, we can talk about all sorts of different geometries, all sorts of different number systems, all sorts of different combinatorial systems, etc. It allows for modeling a great number of smaller formal systems, each with its own characteristics and properties.

Then, there is math as people who want to *use* it see it. For these people, math is a language that is useful for modeling what is observed. And the fact that this language allows for a great variety of patterns means that there is almost always some aspect of mathematics that can model what is observed. Even in the worst case scenario, new math can be invented that helps to describe what is seen.

So, yes, groups were studied by mathematicians long before they were found to be useful for chemistry and physics. But, since groups are, in essence, descriptions of possible symmetries, it is hardly surprising that physical situations involving symmetry can be described by group theory. Similarly, manifolds are a type of 'smooth geometry' that allow a form of calculus to be done, so it is hardly surprising that it is found to be a useful concept when physicists started thinking that maybe the geometry of spacetime is a thing to be studied. Also, don't forget the 'ideas in the air' aspect: if mathematicians have studied it, it is far more likely that a physicist will decide to use that formalization in their own work.

And, of course, when discussing the 'unreasonable effectiveness of mathematics', it is important to acknowledge the wide swaths of mathematics that have NO known connection to any physical theories whatsoever. And, frankly, this is a very large part of mathematics. it is also important to realize that other aspects of mathematics were *invented* to help describe things in science. So it is hardly surprising that these areas are found to be useful.

And, don't forget that because math is incredibly diverse, if one model fails, there is almost always another model close by that will succeed. if flat spacetime doesn't work, use curved; if groups don't work, use monoids; if monoids don't work, use categories, etc.

Plato liked to use math as a key point in his argument that there are 'eternal forms', but I think the revelations of modern math have made that a rather untenable position. We *know* there is more than one possible geometry. We *know* there is more than one possible number system. We *know* that there are other axioms we could use and get much of the same expressiveness. We also *know* that many ideas in math will never have a precise correspondence in the physical world. For example, whether the Continuum Hypothesis is true or not will have no effect on any physical theory I can imagine, but we *know* that we can either assume it to be true OR assume it to be false *and get equally consistent mathematics*.

TL;DR: math is a sport for mathematicians, and a very expressive language for others. It is expressive enough to allow for many different ways to describe almost anything.

---

It's Penrose, so I'm going to take him seriously.

He's a smart guy! 

Here's an article I found just now, by coincidence. It touches on a number of issues related to the topic here, I think. It's particularly interesting to me how Newton changed what science was looking for, by giving up on material causal explanations and looking instead for mathematization. So in a way, science says it has understood the material when it stops looking at the material and changes it into an abstraction -- math. 

https://americanaffairsjournal.org/2020/...t-meaning/

Much of what the writer refers to in passing is described in detail in Burtt's classic book The Metaphysical Foundations of Modern Science. 

https://www.amazon.com/Metaphysical-Foun...oks&sr=1-1

Which may be pirated here:

http://gen.lib.rus.ec/book/index.php?md5...7F3D3AD067

As an early critic of the Enlightenment, William Blake accused Newton of valuing the abstracted understanding over the genuine material -- of claiming that truth is in the formula rather than in the thing itself.

Burtt's book is limited by it focus (necessary at the time of publication) to Newtonian physics. But the metaphysical problems that arose from relativity and quantum mechanics go far beyond those presented by Newtonian physics. That this book was written in 1932, just after the formulation of Schrodinger's equation, is quite enough to question its applicability to modern science.

[Belacqua]

Plato liked to use math as a key point in his argument that there are 'eternal forms', but I think the revelations of modern math have made that a rather untenable position.

The fact that there are different maths that derive from different sets of axioms doesn't affect Plato's thought at all.

math is a sport for mathematicians

You make it sound unserious.

Burtt's book is limited by it focus (necessary at the time of publication) to Newtonian physics. But the metaphysical problems that arose from relativity and quantum mechanics go far beyond those presented by Newtonian physics. That this book was written in 1932, just after the formulation of Schrodinger's equation, is quite enough to question its applicability to modern science.

The book is an accurate history of the changes in scientific epistemology that happened because of Newton. Much of what he says remains the same despite the switch from classical to quantum physics. The fact that science agrees NOT to explain many things, instead accepting them as brute facts ("it happens that way because it just happens that way") and considers them explained when they have been given a math formula remains the same.

Most importantly, the book is important for people who say that science and only science gives us truth. We can see that science is a contingent system -- it could be different -- with a genealogy. The kind of answers it looks for determines what it will and won't find. That means that it leaves huge aspects of the world unexplained and, in the current paradigm, unexplainable.

[Jehanne]

Was watching a debate between Graham Oppy and WLC recently on YouTube, and the topic of the debate had to do with whether or not the applicability of mathematics to the universe serves as evidence for God. WLC (and a lot of Christian thinkers) seem to think that mathematics (at least at the advanced levels) is completely a priori and will find it surprising that it nevertheless can be reliably applied to the universe in the form of physical laws and such. But I'm really not sure what the shocker here is. My understanding is that even the most advanced mathematics that is applied to reality is generally still based on aspects of reality that have been discovered/experienced, so of course when you then apply mathematical conclusions and theorems back to reality, it shouldn't be a surprise that often times there will be a successful mapping between mathematics and reality. If there is structural order in the universe, this is to be expected. But order does not necessitate the existence of God and is perfectly compatible with naturalistic views. Order, for example, might simply be the necessary manifestation that the universe exhibits, and pure chaos might perhaps be some form of an illusion. You could argue that this order still needs some grounding in something, but that is still fine with naturalism in general.

Thoughts, disagreements, go ahead.

Craig has abandoned that so-called line of "reasoning":

RationalWiki -- William Lane Craig

[Cepheus Ace]

listen here my young apprentice, what WLC is doing is pushing for platonic realism and substance dualism in order to sneak in the idea that his god exists without evidence

This is why it's far better to ignore WLC and read the good people who raise the same issue.

WLC also argues against platonic realism in his writings but he shamelessly uses it when it comes the kalam cosmological argument.

christian apologists are shameless frauds at best

[GrandizerII]

Was watching a debate between Graham Oppy and WLC recently on YouTube, and the topic of the debate had to do with whether or not the applicability of mathematics to the universe serves as evidence for God. WLC (and a lot of Christian thinkers) seem to think that mathematics (at least at the advanced levels) is completely a priori and will find it surprising that it nevertheless can be reliably applied to the universe in the form of physical laws and such. But I'm really not sure what the shocker here is. My understanding is that even the most advanced mathematics that is applied to reality is generally still based on aspects of reality that have been discovered/experienced, so of course when you then apply mathematical conclusions and theorems back to reality, it shouldn't be a surprise that often times there will be a successful mapping between mathematics and reality. If there is structural order in the universe, this is to be expected. But order does not necessitate the existence of God and is perfectly compatible with naturalistic views. Order, for example, might simply be the necessary manifestation that the universe exhibits, and pure chaos might perhaps be some form of an illusion. You could argue that this order still needs some grounding in something, but that is still fine with naturalism in general.

Thoughts, disagreements, go ahead.

Craig has abandoned that so-called line of "reasoning":

RationalWiki -- William Lane Craig

Even though WLC is not the main point, the debate I was referring to happened like very recently and he was using this line of reasoning. Do you mean he abandoned it very recently?

---

https://www.youtube.com/watch?v=H9Q6SWcTA9w

Just saw the video. To be honest, he doesn't really say much here. I'll give those other links a check. I do personally agree with polymath's answer, ftr. Example, if there's symmetry involved in the real world, then the maths that makes use of symmetry will naturally be applicable to the real world.

[UtilitarianDeist]

I'm terrible at math but that must mean that God doesn't want me to be good at math so I'm fine with that. God obviously wants plenty of other people to be a good at it.

[Belacqua]

Example, if there's symmetry involved in the real world, then the maths that makes use of symmetry will naturally be applicable to the real world.

I don't think anyone doubts that math has applicability to the material world. We could barely get through a day without it.

The controversy starts up when people say that math is always and only a description of the material. The metaphor that math is only a language to talk about the material appears to break down at some point. 

And it's not only Platonists who say that numbers have a kind of independent existence, in a non-language kind of way. The number 2 exists in a way that the word "cat" doesn't, for example. That's what Popper, Penrose, and many others say is the case. 

In the end it's a metaphysical question, not a scientific one. People who are fully committed to a material-only kind of metaphysics will deny that there is anything else. Since I don't know the answer, and I take Popper and Penrose et.al. seriously, I have to keep an open mind on this.

[The Grand Nudger]

Penrose and Popper, both, were committed and vocal materialists. You're reading what you believe into the things they wrote.

Three worlds is not a dualist conjecture. It's the observation that these three categories - the world as it is, how we experience the world, and what we know about the world, are not equivalent. It's a useful conjecture in that world three, of which science is a representative subject, is not necessarily equivalent to world one, the world as it is. The aim was to improve world three, so that it might be closer to world one.

In that context, it works. However, in the context of mind, it doesn't. Our brains are a representative of world one, our experience is a product of that brain, and what we know (or think we know) is a consequence of that experience. Penrose's contention is not that mind is immaterial in any way shape or form. It's that classical physics and computation might not be able to explain how our brains work, that we need a better understanding of the quantum world for that.

If you insist on making these constant appeals to authority, then it might help to pick an authority that actually does express the argument that you're trying to make.

[polymath257]

Example, if there's symmetry involved in the real world, then the maths that makes use of symmetry will naturally be applicable to the real world.

I don't think anyone doubts that math has applicability to the material world. We could barely get through a day without it.

The controversy starts up when people say that math is always and only a description of the material. The metaphor that math is only a language to talk about the material appears to break down at some point. 

And it's not only Platonists who say that numbers have a kind of independent existence, in a non-language kind of way. The number 2 exists in a way that the word "cat" doesn't, for example. That's what Popper, Penrose, and many others say is the case. 

In the end it's a metaphysical question, not a scientific one. People who are fully committed to a material-only kind of metaphysics will deny that there is anything else. Since I don't know the answer, and I take Popper and Penrose et.al. seriously, I have to keep an open mind on this.

Yes, there is an aspect of math that cuts across cultures. But this is also true of other basic linguistic concepts. So, cat, chat, gato, mao, etc as opposed to two, deux, dos, er.

One difference is that math  is a *formal* language: it has internal rules that are not present in most natural languages. And, for mathematicians, playing with and exploiting those formal rules are the essence of the game.

And, yes, mathematics really is like a very complex game for those doing mathematics. It has rules about what 'plays' are legal, it has goals (theorems), etc. It can even be helpful to *think* of the mathematical concepts visually and in other ways.

In exactly what sense do numbers have an 'independent existence'? From what I can see, the 'number 2' is a shorthand for all the cases where counting two objects is a useful thing to do. And the mathematical object 2 allows for such modeling.