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Why is mathematics

  1. Mar 28, 2006 #1
    why is mathematics so effective in describing natural phenomena?

    i cannot understand why would nature obey mathematics? It seems that if i define mathematically, for example, the rate of cooling of an substance, than no matter what mathematical operation i do on the formula (allowed by mathematics) and come to another new mathematical expression, new relationship, i would find nature actually exhibiting the derived behavior.

    How can this be if mathematics are invented by people? Is mathematics some way fundamental to nature? (how and why). I would like to post some detailed question, but im jsut stunt with the fact as i stated it. Is there any theory (philosophical) on this.

    I was thinking about this while reading a quote from einstain, which in my opinion is the most profound thing (obvious, but not realized by many), that "Theory determines what we observe".
     
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  3. Mar 28, 2006 #2
    It is a really fascinating question to ask why nature seems to follow mathmetics, but maybe its mathematics that follows nature. Isn't it us who came up with mathmatics to describe nature?
     
  4. Mar 28, 2006 #3
    Yes, mathematics were invented to describe nature, but the operations in mathematics seem arbitrary. For example, i define math relationship between temperature and pressure. Then independently i define relationship between velocity and mass, and so on. How is it, that i can then substitute into different relationships derivations which initially were not even concerned with temperature? (just an example)

    A history showed that many times a natural phenomenon was discovered only after playing around with mathematical formulas (which is becoming prominent in modern physics). How come integrating/derivating and comming up with new math rules, it always seems that nature follows whatever i do with the equation?

    It seems not real to me. It seem that we only perceive nature (through math) as real. But maybe, aliens somewhere do not use "newton's" laws of gravity and have totally different way of relating things together and get "their" math straight also, hence, they perceive nature (reality?) differently?

    Or am i wrong, maybe we could derive "newton's" laws from their laws and vice versa. Hence no reality? (different reality for different descriptions? at the end there are more than one way to Rome) :d

    what do you think?
     
  5. Mar 28, 2006 #4
    What's even more intriguing is that concepts that originated as forays into mathematical abstraction, made in accordance with mathematical procedures but not intended to have any basis in or connection to physical reality at all, are becoming increasingly relevant to modern physics. Non-Euclidean geometry, for instance, was no more than a mathematical curiosity until Einstein applied it to physics with successful results--so while mathematics in general may "follow nature", it hasn't followed it linearly in many recent cases.
     
  6. Mar 28, 2006 #5
    exactly the point. If mathematics were invented, how come the nature seems to obey it?

    This would go along my thinking, we invent nature and the laws. The laws are nature are pretty arbitrary from our point of view. We could have had different equations for physical laws given that we would have agreed on different math rules (which would be consistent, for example, defining 1+1=4 and extrapolating, we can have all the mathematics we want and self consistent).

    Again a beautiful saying of einstein "A THEORY DETERMINES WHAT WE OBSERVE".
     
  7. Mar 29, 2006 #6
    What's the difference between £$&GYGG<<<>$DMN/&G%RG and E=mc^2 ?
    They are both an arbitrary sequence of symbols, only that the second has a "consequence" for us humans, it maps to a certain way we humans, as an arbitrary stable energy structure, interact with the world. This interaction is important for us since it has consequences in terms of pain and pleasure, even indirectly, since being able to manipulate the world through "scientific" knowledge means that we manipulate it to our own desires and to minimize pain.
    But if we didn't perceive pain/pleasure how would we interact with the world ? What kind of science would we develop ? Or if we were hardwired differently and perceived pain and pleasure through odd connections, wouldn't we develop a completely different science ?

    For example, if we were hardwired in such a way that by manipulating a small peeble on a road we could feel intense pain or pleasure, we would develop a complete science only based on that peeble since that has a direct consequence for us, not because the peeble has anything special about it. It becomes special only through our interaction with it. Science is a function of the way we are hardwired with the world in terms of pain and pleasure. A sequence of cause and effect or any logical construction isn't more or less valuable, special or meaningful than any other sequence of random symbols. Cause and effect, mathematics, logic, language, explanations are just other random sequences of items which become important only to us since it has consequences in terms of how we interact with the world in terms of pain/pleasure.

    But our interaction and the way we are hardwired is ARBITRARY. At a deeper level there is nothing special about our structure, our mind, or our science since it came about by a quirk chance. There can be millions of other very different structures with different minds and sense organs, that interact differently and produce/invent a completely different "science".
     
  8. Mar 29, 2006 #7

    Tom Mattson

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  9. Mar 29, 2006 #8
    Eugene Wigner wrote about "the unreasonable effectiveness of mathematics in the natural sciences'', and Einstein himself wrote that "the most incomprehensible thing about the world is that it is comprehensible''.

    It's an old problem.

    "The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things." (Aristotle, Metaphysics 1-5)

    Plato: the human soul being a portion of the world soul, human reason can partake of divine reason.

    Kepler: man being an image of God, his mind can rethink God's creative thoughts which are mathematical.

    Popper believed that the world of mathematical ideas exists by itself and governs both our minds and the world of matter.

    Darwinians, on the other hand, claim that our mathematical abilities result from an evolutionary adaptation to the world. I, for one, find it hard to swallow that the fantastic agreement of contemporary physics with experiments results from natural selection. If this can be called an adaptation, it is a pre-adaptation, and if pre-adaptation were actually happening, this would explode the myth of natural selection as the sole driving force behind evolution. (No, I'm not a creationist. I'm not even a Christian, nor a Muslim, Jew, Hindu, Buddhist, or what have you.)

    Bertrand Russell thinks that "physics is mathematical not because we know so much about the physical world, but because we know so little; it is only its mathematical properties that we can discover."

    He has a point. It seems to me that with quantum mechanics we have reached the limits of the usefulness of maths. The hard problem of making sense of the quantum formalism cannot be solved by writing down another formula. To advance further, we need philosophy, but unfortunately this isn’t the forte of mathematical physicists. To throw in a quote of Max Born: "It is true that many scientists are not philosophically minded and have hitherto shown much skill and ingenuity but little wisdom."

    Let us also not forget that there are plenty of natural phenomena that cannot be mathematically described, such as consciousness and qualia. To argue that these things are not real (simply because they cannot be mathematically described) returns us to the childish enthusiasm of the Pythagoreans.

    Apropos of your Einstein quote, which taken out of context risks being misunderstood. Heisenberg once told Einstein that what had guided him in his research was Einstein's demand that theories ought to be formulated without reference to unobservable quantities. Einstein replied something to this effect: "Even if I once said so, it is Quatsch (nonsense)." His point was that before one has the right theory, one does not know what is observable and what is not. In this and only this sense did Einstein say that theory determines what we observe.
     
  10. Mar 29, 2006 #9
    I beleive it to be down to the fact that our brains work, like computers, in an inherantly mathematical manner and as such can only understand and formulate patterns and concepts about the world and nature using what is essentially mathematics & logic.
     
  11. Mar 29, 2006 #10
    Then truth is relative to our arbitrary structure. If atoms were made up of small elvis presleys or any other thing, it would not matter, they could be made up of any structure and have any kinds of properties and laws associated with them. That they happen to be what they are, governed by quantum mechanics is just a quirk choice, any other structure would be equivalent. Any universe and laws of physics are based on some kind of regularity, exactly which regularity it happens to be in any particular universe is irrelevant, anything else would do. We study these particular laws because they have direct consequences on us, but if we ended up evolving into another hardwired being we would experience and study maybe completely different laws and never even discover any laws of physics since they would be indifferent to us if we didn't perceive their influence. Even logic and language is just a quirk random combination/choice, anything else would do. Maybe this is where art meets science.
     
  12. Mar 29, 2006 #11

    selfAdjoint

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    I have thought about this over the years too. Here is my idea: Mathematics studies relationships, all kinds of relationships to all different levels of abstraction. They put no restriction on the kind of relationships they study except logical consistency.

    Now any universe has in it SOME relationships and it is highly likely that they and their interconnections are described by some part of existing mathematics. If not, the mathematicians will eagerly fall on the new relationships and erect a new part of their study around them. They did this with the Dirac delta "function" and with Witten's superstring monster math.

    But it is not true that nature is described "by mathematics"; it is described by "just a little part of mathematics". There are vast areas of mathematics that so far, seem to have no relation at all to nature.
     
  13. Mar 29, 2006 #12
    Not even logical consistency is necessary. Any universe with any structure, even contradictory, even a random sequence of symbols-concepts-contradictions can describe or be the substrate of a random arbitrary universe.

    To be more precise science is also carved out from, aside from pain/pleasure, other more abstract emotions/feelings such as "good and bad" or "satisfying and non satisfying" or "coherent and non coherent" etc.

    The terms are that these emotional states carve out how we follow the path of knowledge/science. It implies some kind of abstract "judgment" as a starting point. This starting assumption of judging according to how knowledge rings with our emotions, which means how it is useful to us or valid "culturally" is ARBITRARY. It is well defined within our interactions with the wolrd and other humans, but from a more abstract point of view, from outside the system, it is just as valid as any other random sequence of items/symbols/concepts or inventions. If our mind/emotions were hardwired differently, random stones could be the most important thing for our mind/emotions as opposed to atoms for example.

    Man, in this sense is the measure of all things, but there are many infinite more things that Man doesn't measure or consider.
     
  14. Mar 29, 2006 #13
    yes if we specifically look and apply math to some specific relationship. But how come by doing equation manipulations, substitutions, integrations/derivatives etc nature always seems to "obey"?

    how come finding relationships between numbers can be applied to nature? It seems that nature tries to by "logical" (mathematically). For example, i have to invent math (for ex. Lee algebra) to solve some problem of nature. While inventing relationships between numbers and borrowing mathematics of earlier days, i am able to define set of mathematical rules. I can do with those natural variable any manipulation of the mathematically defined rules and it seem to work in nature just as i described in mathematics. Its fascinating to me :surprised

    IS there any possible connection of math to nature? Or rather the other way around, is there possible other "language", other than mathematics, that can manipulate relationships, which when invented/discovered would be even better than math to describe nature?
     
  15. Mar 29, 2006 #14
    Mathematics is applied to the real world if you assign a physical entity a number and compare it to other numbers which are assigned to other entities.

    It is 45 degrees now. What does that tell you?

    If you look carefully at how math is applied to physics you find that every single formula has limits, but we don't talk about them and even ignore them. For intance, $d = v*t$ , is valid if $v << c$

    why do we have to restrict domains of every formula in physics? What happens outside of the domans. Like $v >> c$ ?

    An ultimate test of mathematics would be to predict the fundamental constants G, h, c, e, etc, and masses of all particles.

    A theory of everything could do that, but it would most likely be a subset of all mathematics.
     
  16. Mar 30, 2006 #15

    mathwonk

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    its easy - god is a mathematician.
     
  17. Mar 30, 2006 #16
    Could nature not be mathematical ? Can we imagine any kind of relationship between items that is NOT mathematical ? No, we can't, even the most confusing - contradictory - random relation or structure can be broken down into some kinds of mathematical items. Natural language itself is just a sophisticated form of mathematics, like a descriptional language that can be ultimately reduced to some kinds of mathematical items and relationships.

    Also our sense organs don't do anything but provide a very large flux of NUMBERS to our brains which then convert and manipulate them mathematically. Imagine our sense organs receiving analog electrical signals, then convert them to digital and hence all reality is a sequence of numbers.
    Like a film on DVD is 10 billion bytes yet decoded correctly you get a 2 hour film.
     
  18. Mar 30, 2006 #17
    Math is truth.


    It's the only thing that one can prove in this universe.

    I remember reading some guy's lucid dream about meeting God. In it, he asked what the meaning of life was. What was returned was a calculation that endlessly expanded and increased in complexity. This is just his interpretation of what it would be, but I don't think it's far off. Millenia from now, when people are assimilated with technology to enable them to calculate things which we cannot imagine, I believe one would be able to predict actions which we think are random.
     
  19. Mar 30, 2006 #18

    selfAdjoint

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    Wrong. It's true that math is the only field where proof means anything, but those mathematical proofs don't start from bedrock; they start from assumed axioms. Now the lowest level axioms have to sound "reasonable" but people can and do differ on even the axioms of set theory. So mathematics just does not arrive at big-T Truth, it arrives at valid conclusions from its assumptions.
     
  20. Mar 30, 2006 #19

    Hootenanny

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    I'm afraid you can never prove anything, only disprove it. As for mathematics, in my opinion it is just 'our' interpretation of the world, it is a set of arbitrary values we assign to external stimuli in a logical fashion in order to describe our experiences.
     
  21. Mar 30, 2006 #20
    Is then mathematics subset of logic? Its amazing that language of mathematics not only describes but also at the same time incorporates logic in it. The keyword is describe! SOmeone (us) have to explain (interpret) what the description is, right? (even though i cannot draw a line where description meets explanation :shy: )

    how do you know the manipulation inside our brain is mathematical nature? Does the brain take curl of E to get light? :rofl: It also implies our brains are computers with neurons somehow arranged into logic gateways which mechanically crunch data and send output to "somewhere in brain". However, somewhere there must be interpretation of the output.

    Maybe, our being is not brain vs inputs. maybe we are all integral part of our environment and each and every "particle"/object has degree of consciousness on its own. We see it in quantum that electrons seem to make "decisions". We perceive ourself as distinct objects, however, this is only because of reality of our perceptions. If we could see only waves we would not see, table, chair, ourselves ...., but rather just mesh of waves which are all interconnected. Therefore, fundamental is level of abstraction? At some level we are just "energy" acting in empty space. (forces between quarks, electrons, etc)

    it seems im just braging now :eek:
     
  22. Mar 31, 2006 #21
    Any manipulation inside the brain will always be mathematical-logical. As I said, "Natural Language" itself is a pure logical-mathematical language only it uses very high abstractions and rules of engagement.

    Are the ultimate elementary particles of reality simply numbers ? When everything is reduced to their fundamental equations we have relationships and numbers (think quantum electrodynamics). A bit like a simulation of reality on a computer that can be reduced to bits, and in fact there are theories that state the "it from bit" concept of physics-reality in that all is information. We may be even able to produce a "transcendental matter" within computers composed of many more elaborate and complex relationships than quantum mechanics that govern atoms and create completely new - exotic virtual atoms. In future giga computers we may truly have completely new kinds of "matter" with many trillions of levels in the reductionist sense or not even reductionist anymore as the larger can be "contained in the smaller" and even more odd relations.

    Maybe the mind is the ultimate elementary particle, maybe the mind cannot be reduced into anything and is the starting point and ending point of reality. This "olistic" theory has also been around some time.
     
  23. Mar 31, 2006 #22
    From a pragmatic standpoint, logic (and by extension mathematics) is simply the best system mankind has concocted so far to distinguish bad arguments from good arguments. Given its basis in evidence to make arguments, one could also argue that it has some empirical basis, though higher-level mathematics has divorced the system of logic from its vaguely empirical basis to such a degree that the extent to which this is true is probably minimal at best--though it is again interesting to note cases in which mathematical trinkets were created with no physical references intended or in mind, only to be successfully applied to the natural world later.

    Even when it does have an obviously empirical basis, logic is an imposed structure by which one can make inferences in order to draw a conclusion, and to what extent the rules of logic are utterly arbitrary is difficult to determine.
     
  24. Mar 31, 2006 #23
    Thinking about reductionism, it may be that there is no absolute size. The universe may be made in such a way that as we get to smaller and smaller items at a certain point we get to (maybe at 10^-100 cm) back to the universe all over again. Like if we look inside an imaginary microscope and get past a small enough size limit we start to see our whole universe all over again. Then going smaller we see our galaxy, sun, earth, ourselves all the way down again recursively forever. At that given size limit where the smallest item corresponds to the entire universe is where "meatphysical entities" lie, or our mind, or minds or god's mind or mathematical objects lie. That ultimate item doesn't respect mathematics in the sense that the smallest size is equal to the largest, so it lies outside this universe and logic and yet it is the interface between mind and matter. A mathematics where 3 is equal to 3^100 may even be possible to construct.

    The point is that if size is infinitely recursive in this sense, then there is no absolute size or measurement, they are only relative to each other within a certain range. The universe would seem to lie outside of any mathematical description or maybe even any logical description. We have no extension and our mind is the ULTIMATE ELEMENTARY PARTICLE AND COMPLETE UNIVERSE AT THE SAME TIME.
     
  25. Apr 1, 2006 #24
    One way to imagine a universe with all items having a "constant size" is imagining them decreasing their size in some dimensions of an n-dimensional space and increasing their extension in the hidden dimensions such as their volume is always constant. So the item of size 10^-100 cm will have a length in the hidden dimension of 10^100 cm so as the total volume is always constant. And when the size reaches the "mind as elementary particle" limit it just switches dimensions and starts all over again. I mean a universe and mathematics where the electron (or planck size item) is equal in size to the universe is conceivable and developable, surely by much smarter mathematicians then myself.
     
  26. Apr 1, 2006 #25
    Interesting. Then what would happen if the universe was an INFINITE DIMENSIONAL UNIVERSE ? Then the big bang could be seen as some small items in a set of hidden dimensions reaching their limits and all of a sudden start expanding in our 3 dimensional universe. Or maybe the dimensions are constantly being created to accomodate ever expanding sizes of items. Or maybe at certain size limits new dimensions come into being to keep the volumes constant.

    Maybe the extra dimensions introduced in string theories and such should be gigantic dimensions instead of microscopic dimensions. An infinite dimensional universe is even more odd then a universe that has small sizes equal to large sizes.

    Anyways the fact that even at the smallest limits everything can be described mathematically implies that even at macroscopic sizes everything follows some logic-mathematics no matter how hard it is to figure out.
     
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