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Why does math work in our reality?

  1. Sep 3, 2009 #1
    I’m reviewing my mathematics knowledge, except I’m looking for a different reason. I understand how it works you know, 1 plus 1 so on, I’m trying to understand why it works.

    Pure and applied mathematicians and physicists have tied our understanding of reality with mathematics. They believe that if it computes it is real. This will do for a while, till we pose a question beyond our understanding.

    But that’s for a different time. Why does it work?

    Any takers?
     
  2. jcsd
  3. Sep 3, 2009 #2
    Hi there,

    You have a different uderstanding of mathematics applications than mine. I don't see mathematics as an end solution to undrstand different real situations.

    No matter what happens, with or without scientists, the world keeps on turning, and objects keep on falling when dropped. Therefore, I see science as a way to MODEL mathematically certain real life situations. The models can be right on the dot or far away from the truth, but the idea is to try to explain certain situation through mathematical models. Therefore, these models are made up to work in this world.

    Take again falling objects. To my knowledge, it never happened that an object started floating when dropped. In addition, it is simple to see that the speed of the falling object increases. Therefore, scientists worked up a mathematical model that tries to explains this situation, anyways that tries to get as close as possible to reality.

    Cheers
     
  4. Sep 3, 2009 #3

    arildno

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  5. Sep 3, 2009 #4

    HallsofIvy

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    This depends strongly on what you mean by "works". NO mathematics exactly fits a physical situation ("reality" if you like). All "reality" involves measurements that are approximate so the best we can hope for from a mathematical theory is that it work approximately.

    All mathematical structures (theories) are "templates". Every mathematical structure involves "undefined terms", words that are defined using those undefined terms, axioms, and theorems proved from those axioms. To apply a mathematical structure to "reality", we have to assign meanings to those undefined terms. IF the axioms are true with those meanings assigned, then all theorems proved from those axioms are true and all methods of solving problems derived from those theorems will work.
    Of course, the axioms won't[b\] be perfectly true, only approximately true. The key to the "Unreasonable Effectiveness of Mathematics in the Natural Sciences" is having a large array of mathematical structures and choosing the one that best fits the specific application.
     
  6. Sep 3, 2009 #5
    But does that mean that in the future the same can be said? That presently we just don't have a full understanding of mathematics.

    I've been wondering this myself. It would've been a strange feeling for the first person who realized early on that mathematics actually had a use in real life. Its almost like the matrix, that behind the veil of reality its just numbers on paper.
     
  7. Sep 3, 2009 #6

    Mentallic

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    It's not our understanding of mathematics that is flawed, it's the models we use to represent reality. Approximations will be made in our calculations of the real world for now, and a long time to come. There will never be exact solutions to reality, except approximations of desired accuracy (futuristic super computers taking all possible variables into consideration in its calculations).
     
  8. Sep 3, 2009 #7
    Hard work.

    Over centuries mathematicians have worked hard to create a model of reality using symbols and logic. Its been refined and tweaked to help us predict the world. Millions of hours of work across multiple cultures, trial and error.

    Might as well ask how can google map work, how can it show us how to get from Los Angeles to New York. Its impressive, for sure. Its just colors and lines on a computer screen. How does it know? It was designed to resemble the larger scale, in a very intentional and logical way.

    There is no miracle or perfection to it. But of course, its been in the making quite a while, so for some it seems like magic. The pyramids weren't built in a day, or by one person, and they are an awesome site to behold. But people build them, one stone at a time, over many years.
     
  9. Sep 3, 2009 #8

    I'd say there was probably no other way to construct an orderly & comprehensible and consistent universe. What other way is there for a universe like ours to exist, in which it would not be reigned by total chaos and where particles would not have random values and no physical law would be possible?

    How would there be laws of physics if there was no math? And how would there be you, if there were no laws of physics?

    You want a world without math and laws of physics? You are not talking of a world, you are talking about an Idea.

    The universe can be classically thought of as an aquarium, where we are the marine species that got smarter.
     
    Last edited: Sep 3, 2009
  10. Sep 3, 2009 #9
    Math probably "works" in reality because math always works. Math is nothing but what we can logically deduce about fundamental abstract ideas such as integers and sets. The space of logically-consistent ideas is much, much larger than what is real. The universe just one idea in an infinite sea of other things which are just as feasible.

    Mathematics seems to correspond very well to reality, but that's only an evolutionary result. The best results all stem from the geometry of the world we find ourselves in. It's like in literature, how some people claim that all modern works are a retelling of something found in a Shakespearean play, all mathematics leads back in some form to ideas understood by Euler. Exponentiation, prime numbers, and bell curves are all relatively new ideas, but they all have some relation to the same circle understood 2,000 years ago by Pythagoras.
     
  11. Sep 4, 2009 #10
    To echo Kant: How could it not work? Math is a product of our perception of the world and our ability to reason within it. Reason is shaped by our perception of reality, not the other way around. In other words, its not reality that is adapted to mathematics, but rather its our minds, and consequently mathematics, that are adapted to reality. Math is an inseparable part of the way we see and experience the world: 2 + 2 = 4 is not something we can conceive of as being false. More generally, we can't conceive of a world where math does not work.
     
    Last edited: Sep 4, 2009
  12. Sep 4, 2009 #11

    arildno

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    "2 + 2 = 4 is not something we can conceive of as being false. "

    Sure we can.

    Argument, please?
     
  13. Sep 4, 2009 #12
    1+1=2 only because it was defined that way.

    One could just as easily define a different set of symbols for numbers and operators so that 1+1 = 1. All of these symbolic choices were entirely arbitrary.

    Everything in math results from a few base assumptions of countability that mirror the world
     
  14. Sep 4, 2009 #13
    Really? Then I imagine you are quite comfortable with square circles and flat balls.

    By 2 + 2 = 4, I mean the straightforward, everyday meaning of the statement: "Two and two objects put together, make four objects". If you begin to speak about a different "set of symbols" and "operators" you are in an arena far away from the intended context.

    This is like objecting to "I am now writing on physicsforums" because the words "I", "am", "now" etc. have entirely arbitrary meanings and "one could just as easily define a different language so that this sentence is false". Yes, of course.
     
  15. Sep 4, 2009 #14

    arildno

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    " I mean the straightforward, everyday meaning of the statement"

    What relevance does this have to the validity of mathematical statements??
     
  16. Sep 4, 2009 #15
    The case may be made clearer by considering a genuinely empirical generalization, such as 'All men are mortal.' It is plain that we believe this proposition, in the first place, because there is no known instance of men living beyond a certain age, and in the second place because there seem to be physiological grounds for thinking that an organism such as a man's body must sooner or later wear out. Neglecting the second ground, and considering merely our experience of men's mortality, it is plain that we should not be content with one quite clearly understood instance of a man dying, whereas, in the case of 'two and two are four', one instance does suffice, when carefully considered, to persuade us that the same must happen in any other instance. Also we can be forced to admit, on reflection, that there may be some doubt, however slight, as to whether all men are mortal. This may be made plain by the attempt to imagine two different worlds, in one of which there are men who are not mortal, while in the other two and two make five. When Swift invites us to consider the race of Struldbugs who never die, we are able to acquiesce in imagination. But a world where two and two make five seems quite on a different level. We feel that such a world, if there were one, would upset the whole fabric of our knowledge and reduce us to utter doubt.
    ...
    A similar argument applies to any other a priori judgement. When we judge that two and two are four, we are not making a judgement about our thoughts, but about all actual or possible couples. The fact that our minds are so constituted as to believe that two and two are four, though it is true, is emphatically not what we assert when we assert that two and two are four. And no fact about the constitution of our minds could make it true that two and two are four. Thus our a priori knowledge, if it is not erroneous, is not merely knowledge about the constitution of our minds, but is applicable to whatever the world may contain, both what is mental and what is non-mental.
    ...
    Let us revert to the proposition 'two and two are four'. It is fairly obvious, in view of what has been said, that this proposition states a relation between the universal 'two' and the universal 'four'. This suggests a proposition which we shall now endeavour to establish: namely, All a priori knowledge deals exclusively with the relations of universals. This proposition is of great importance, and goes a long way towards solving our previous difficulties concerning a priori knowledge.
    ...
    In the special case of 'two and two are four', even when we interpret it as meaning 'any collection formed of two twos is a collection of four', it is plain that we can understand the proposition, i.e. we can see what it is that it asserts, as soon as we know what is meant by 'collection' and 'two' and 'four' . It is quite unnecessary to know all the couples in the world: if it were necessary, obviously we could never understand the proposition, since the couples are infinitely numerous and therefore cannot all be known to us.

    http://www.ditext.com/russell/russell.html

    See also the Kant / Math thread. I would be interested in seeing a counter-argument from a logician. One probably exists somewhere.
     
  17. Sep 4, 2009 #16
    Explain. Of what type of validity are you talking about?
     
  18. Sep 4, 2009 #17
    Mathematics is the study of structure, rythim, and patterns.

    The Universe has structure, rythim, and patterns.
    Therefore, math works in our reality.
     
  19. Sep 4, 2009 #18
    See Frege's Theorem: http://plato.stanford.edu/entries/frege-logic/. Arithmetic follows from the validity of second-order logic and Hume's Principle. Are you denying the validity of second-order logic, or are you denying that "for any concepts F and G, the number of F-things is equal to the number G-things if and only if there is a one-to-one correspondence between the F-things and the G-things?" I suppose you could also be denying the validity of the proof itself...
     
  20. Sep 4, 2009 #19
    In that case, the reason that math works in real life can simply be stated as "conservation of energy" (from physics). Conservation of energy which is generally true at a macroscopic level allows things to be accumulated, and this allows us to define symbols for countable quantities, and everything else basically comes from that axiom.
     
  21. Sep 4, 2009 #20
    How do you explain the fact that a line a circle intersect at most at two points by "conservation of energy"?
     
  22. Sep 4, 2009 #21
    Conservation of energy is why numbers have meaning in our reality. In order to talk about geometry you also need dimensions...and our reality has 3 spatial dimensions, so that gives meaning to 3 dimensional spaces.

    A line intersects a circle in at most 2 points because we arbitrarily defined the concepts of "lines" and "circles" to behave that way: a circle is the set of points of constant radius from the centroid, a line is the set of points that can be represented as a linear combination of two points. To have a line intersect a circle in more than 2 points would contradict either the definition of a circle or the definition of a line, and therefore cannot be.

    Because there is a real world analog for vector spaces of dimension less than or equal to 3 (namely, the space-like dimensions of our universe), the statement holds in the real world as well as the abstract made up world of math.

    Of course, this is not coincidental -- everything in math was simply developed so that it could be used as a way to model reality. If there were another universe/reality in which conservation of energy did not hold, then the concept of a "number" wouldnt make sense at all...yet there might be other rules that governed their universe leading to a form of math that was completely absent of the concept of numbers
     
  23. Sep 4, 2009 #22
    Junglebeast, the closest description to what you are saying that I can find in http://plato.stanford.edu/entries/philosophy-mathematics/ seems to be nominalist scientific reconstruction. This idea has largely been dismissed by philosophers, as discussed on that page. Is this correct, or would one of the other described theories better fit your view? Does your view fall outside of the usual classifications of foundational mathematical theories? Has this article missed a theory that you think should have been included?

    In a nominalist reconstruction of mathematics, concrete entities will have to play the role that abstract entities play in platonistic accounts of mathematics. But here a problem arises. Already Hilbert observed that, given the discretization of nature in quantum mechanics, the natural sciences may in the end claim that there are only finitely many concrete entities (Hilbert 1925). Yet it seems that we would need infinitely many of them to play the role of the natural numbers — never mind the real numbers. Where does the nominalist find the required collection of concrete entities?

    The argument presented above is similar to the one given by Russell that I previously quoted. Platonism and the four schools etc all hold math as existing or subsisting in some sort of idealized or abstract way, and not as being purely derivable from or existing in the physical realm. It is also possible, of course, that I am misreading these descriptions, in which case I would appreciate having that pointed out as well.

    ...mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories.
     
    Last edited: Sep 4, 2009
  24. Sep 4, 2009 #23
    kote,

    I don't have the time or interest to read the past philosophical musings of every 19th century philosopher. It does not appear that what I'm saying can exactly be summarized by any of those categories. I also don't believe what I'm saying is a philosophical opinion at all, I think I'm just stating facts.

    Math is simply a logical formalization of something -- anything. There are basic definitions and axioms, and on top of that everything else is derived from those fundamental axioms using logical proofs.

    One can define ANY set of basic definitions and axioms and the resulting set of logical conclusions would be a body of mathematics. However, since we are interested in using mathematics to model reality, all the fields of mathematics are designed with basic assumptions that in some simplified way represent an aspect of reality.

    The answer to the OPs question of "why does math work in reality" is really an ill-posed question because math doesn't actually work in reality. Math simply describes a space of conclusions that can be drawn from a set of assumptions. When you want to apply math to reality, you have to choose an appropriate set of assumptions that mimic the aspect of reality you are interested in modeling...and this usually implies choosing a physics model.

    Even when you are talking about a question so basic as: "John has 1 apple. Beth gives John another apple. How many apples does John have?" which is translated into "1+1=2", this is still assuming a physics model, and that's why it makes sense in real life. You're assuming that there is conservation of energy, and conservation of momentum, among other things...because Apples aren't spontaneously appearing or disappearing. If there were a different alien reality where these principles did not hold, and you posed this question, they might say: You idiot! 1+1 = ?, because apples spontaneously appear and disappear at will!

    One could define any laws of physics that they wanted, and as long as they don't contradict, you could build a field of math around those made up laws of physics. To the degree that the physics is accurate, the mathematical conclusions will be....although the word "degree" may be misleading in this context because obviously a small error can be magnified into a huge error in the prediction.
     
  25. Sep 6, 2009 #24
    Mathematics works in describing the "physical world" because we constructed it in that way.
    Ok, think about this in terms of this analogy: the world (the universe), a painter (scientists), and a painting of a landscape (applied mathematics).
     
  26. Sep 6, 2009 #25

    The analogy will hold, if the painter drew an animation and not a picture. That animation also has to be consistent for 14 billion years into the past, so it's not trivial.
     
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