The Unreasonable Effectiveness of Mathematics in the Natural Sciences

[Steve Smith]

I fear the below continues to be argumentative on my part.   I can't seem to quite discover or render the kernel of my point here.   Apologies to all for my lack of clarity, it is probably from trying to hold materialist,
- Steve

Russ Abbott wrote:

Awhile ago on this thread I said that mathematics is effective because there are regularities in nature. No one commented on that.

I think we all share this as a working understanding of how mathematics and nature relate to each other.   I believe we developed mathematics precisely for that purpose.  One could say our mathematics evolved, others could say we "discovered" it.  Even the more obscure branches of pure/abstract Math seems to have second-order relations... even if it doesn't describe patterns in the world, it describes patterns in mathematics itself (abstract algebra, group/set/catagory theory, etc.)

There have been many comments discussing whether what goes on in our minds matters, but very little about what goes on outside our minds.

My position (and I will look for some references to back it up) is that I don't believe we can *know* that anything exists outside of our mind, that the physical universe, that physical reality is anything more than an hallucination or a projection of the mind.   Naturally even this description is problematic as it begs the question (some more) of "what is mind?" and "who am I chattering on with here, if nothing outside my mind exists, anyway?"

It's amusing to poke fun at the way some people think, but I'm not sure it gets us anywhere.

If you are referring to my references to folk knowledge, quite the contrary.  I hold folk knowledge in high esteem and practice it as a matter of course, possibly more directly than most folks on this list.  It is how I cook, build, grow, heat my house, etc.  all by colloquial understandings of the phenomenology of normal everyday things (foods, fire, materials, plants, etc.)

My curious mind constantly checks these things against the scientific abstractions I know of, and am amused at how well they work and then amused even more when they don't work so well.  I don't imagine for a second that the abstractions are *wrong*, but rather that my observations or application of them is more likely flawed.   For example, when the tree I might be cutting falls a different direction than I intended (and predicted), I don't imagine that gravity hiccuped or even that the "wood sprites" niggled it just to frustrate me.   When the cake I'm baking falls, it is the result of "opening the door when I shouldn't", not the kitchen sprites, but not precisely "pressure waves induced by the ...." either.   On the other hand, an animist understanding would probably work generally as well.  In all cases, I hold (at least) a dual state of understanding...  a colloquial one (the sun rises, passes through the zenith, then sets every day) and a more scientific one (the earth rotates, exposing the sun at different angles through the day) with no problems.  I could maybe even pack in a deist, animist or anthropomorphic understanding on top of that (Ur the Sun God, or blazing chariots, etc.) if it suited me.  And I wouldn't *have to* be confused about which type of understanding I was using, though I certainly could be.

Are there regularities in nature? If so, then why is it surprising that mathematics is useful for describing them?

It would appear that if there is a physical universe distinct from our minds, then there are many regularities and that this thing we call mathematics (which we developed, evolved, or discovered) is specifically useful for describing it.

If there is no such thing, and the physical universe is a projection or hallucination of our mind, then it is even *less* surprising  that our minds have a language specifically attuned to describing it (i.e. Mathematics).

The limits to the former are interesting (hidden variables, dark matter/energy, unified this-n-that theories) and even moreso in the latter (e.g. godel's incompleteness theorem) perhaps.

On the other hand, one might claim that even asking that question is imposing our (perhaps foolish) mental model of what we mean by regularities on nature. But taking that stance suggests that we can't get out of our minds at all and there is no point in having this discussion. 

I concede that this discussion is only interesting to the extent that we accept a separate physical universe as a working model.  We don't have to believe in it in any "absolute" sense, but if it is not our working model, the discussion is generally moot (or masturbatory?).

So which side are you on

All of them.  It depends.

it useful to share with each other what goes on in our (separate) minds?

If such things (separate minds) exist, then yes, it is useful (or at least highly compelling, else why this list, why this discussion?).

Is it possible that what goes on in our minds can be mapped onto what goes on in nature?

Back to the fundamental question.  If the Universe IS Mathematics, then somehow we are saying the inverse... "what goes on in nature" is mapped implicitly onto our minds (or somesuch).   If the Mathematics is *merely* a good language for describing what we observe in nature, then we are doing precisely what you say "mapping what goes on in our minds - Mathematics - onto what we observe in nature". 

I think the above is the kernel of the question/conversation here...  and it does not preclude the question of whether we "invent" or "discover" Mathematics.   Though I've come to prefer "evolve" where the fitness function is (almost) precisely "how well does it predict/describe/explain Nature?"  

I don't suppose many of us would ask "Isn't it amazing how well a beaver is adapted to living in a stream in forested woodlands?"  Intelligent Designers have a simple answer to this, and Evolutionists have a completely different but in it's own way even more parsimonious and compelling answer.   But neither (I don't think) would want to suggest that  "Woodland Streams *are* Beaverness", though I think there is something like an animist understanding that would say that, and to the extent that woodland streams *are* co-evolved with Beavers, such might hold yet-more truth than the first two ("God made it so!"  "Darwin already explained all that!").

So, maybe a different kind of answer would be "Humans and their Mathematical/Logical understanding of the Physical World have co-evolved".  

Or is there no point in attempting to exchange thoughts since they are all just internal foolishness?  Evolution suggests that it is not all just internal foolishness. If it were we wouldn't have evolved to have these thoughts.  One could argue that that thought itself is just as much internal foolishness as any other. But then why bother to write it down and send it to this list?

This sounds like internal foolishness and I can't imagine why anyone would bother to write it down and send it to the list. <grin>

Seriously.   I clearly believe it is important to make the distinction between a (non)provable, absolute truth (The Universe IS Mathematics) and a practical, working model that Mathematics is very useful for describing the Physical Universe.   Science is about measurement, about hypothesis generation and testing, and about repeatability.  It is grounded in the idea of a separate, independent-of-mind physical Universe.    I don't think this understanding, however, precludes the possibility that what *appears* to our minds as an independent, separately realized, physical Universe, is "merely" a projection of the mind (whatever that is).

I've probably only managed to continue to muddy the water here.  Probably from holding too many disparate types of understanding (materialist/existentialist/animist/deist) in my head at one time.  Me and the Red Queen.  Perhaps if I only run faster and think more impossible things before breakfast!

- Steve

-- Russ

On Tue, Apr 28, 2009 at 11:16 AM, Steve Smith <[hidden email]> wrote:

Very well said, methinks. 

An approach needn't even lose it's utility to poke fun at it, it merely has to  lose "Universal Utility".   I believe many folk remedies, crafts, knowledge fall into that category.   They become "vestigal" knowledge for entire generations until circumstances drift far enough (or abruptly enough) that they become the only or best (known) answer to a given problem (again).

Come the revolution, we'll all be chewing willow bark and slippery elm to relieve what ails us, and laughing at our forefathers who thought all medicine had to be manufactured and shipped in a bottle.  In the meantime, such remedies seem somewhere between "quaint" and "absurd".


glen e. p. ropella wroteth circa early c21:

Thus spake Steve Smith circa 04/26/2009 06:06 PM:
  

Nicholas Thompson wrote:
    

Why should nature give a fig for the tricks we play with our words?
      

The Anthropic Principle might have  a play in this.
    

I think this is the fundamental reason for the unreasonableness.  Math,
like any other language, helps us be goal-oriented.  And anything that
helps us be goal-oriented will _seem_ true to us, regardless of whether
it is true or not.

This is the situation for just about any method: burning witches,
hunting Communists, making marijuana illegal, worshiping mythical
beings, meditating surrounded by crystals and incense, voodoo dolls,
murdering people in foreign lands, torturing enemy combatants, etc.  If
it focuses our attention and allows us to maintain focus on some
objective, then, as a tool, it _is_ useful and will _seem_ true.

When it ceases to be useful, we will be surprised, sit back, and wonder
why we were so enamored with it before... and many of us will even poke
fun of and deride those people who still find it useful.

  

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FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

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FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org

[glen e. p. ropella-2]

Steve Smith wrote:
> Russ Abbott wrote:
>> Awhile ago on this thread I said that mathematics is effective because
>> there are regularities in nature. No one commented on that.

Well, sometimes people miss postings and are too swamped by work/life to
respond.  I suspect that was why I didn't comment... I'm usually a sure
bite for such bait. ;-)

> Seriously.   I clearly believe it is important to make the distinction
> between a (non)provable, absolute truth (The Universe IS Mathematics)
> and a practical, working model that Mathematics is very useful for
> describing the Physical Universe.   Science is about measurement,
> about hypothesis generation and testing, and about repeatability.  It
> is grounded in the idea of a separate, independent-of-mind physical
> Universe.    I don't think this understanding, however, precludes the
> possibility that what *appears* to our minds as an independent,
> separately realized, physical Universe, is "merely" a projection of the
> mind (whatever that is).

I think this can be fortified to a stronger statement.  I tend to
subscribe to the idea that reality is constructed by (and constructs)
that which composes it ("impredicativity").  Hence, there is no "out
there" in isolation from "in here", just as there is no "in here" in
isolation from "out there".  Hence, there is no absolute truth and,
likewise solipsism is false.

Although we _think_ science is grounded in the idea of an "out there", I
speculate that it's not.  Science is grounded in _behavior_, not thought
or knowledge.  Any ideas we may entertain as a result of the behaviors,
the (transpersonal) methods, is an abstract (and therefore inaccurate)
conception of objects, stuff, ... but it's really a tightly knotted
holarchy of stuff and process.  And the stuff, as well as the behaviors
are transient.

Hence, the _value_ we find in "sharing" our ideas doesn't really lie in
how accurately they describe reality.  The value lies in the fact that
the more _attention_ we can recruit to a given perspective or aspect,
the more we can construct the world to be the way we want it to be.

If everyone thinks wildly differently, then the network is too loose and
we swim in chaos.  If everyone thinks the same, then the network
collapses to a single node and we freeze in order.  But with the right
balance of homo- and hetero-geneous attention, we can manipulate the
universe in appreciable ways.

At the end of the day, a good maxim is the ancient aphorism: "Know ten
things, say nine."

--
glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com


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