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Faith and Mathematics?

  1. Jul 7, 2013 #1
    I apologize if there's any mistake in my question or argument as I don't have an advanced education in mathematics (haven't seen anything beyond first year calculus with the exception of partial derivatives in a purely thermodynamic context) so please do correct me if I'm wrong.

    I can't quite wrap my around a few things in math.

    Is math true? If it is something that is constantly discovered by intelligent beings around the Universe then why is there no empirical basis for Mathematics. From my perspective, math seems to be based on intuition as it defines itself. An intuitive basis on the validity of a conjecture (I suppose in this case an axiom) is generally poor proof or a poor starting point in the study of the natural/physical sciences.
    To give an example, an intuitive statement on a natural phenomenon would be "organisms are <i>designed</i> because of the specificity of their anatomy and behaviour to their lifestyle or environment".

    Is math man-made and completely subjective like painting or writing? This would imply that math has no basis in the physical/natural world but (as Wigner would say) "Math is unreasonably effective in the natural sciences." Math can potentially arise in a light of an observation/question (Newton's invention of calculus to determine instantaneous velocities) but there are many cases where math seems to exist on the sole foundation of math itself and has no known application or basis in the physical world. What's more unfathomable is that this math with no known application is suddenly an amazing descriptor of the behaviour of some newly discovered particle or something of that sort.
     
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  3. Jul 7, 2013 #2

    WannabeNewton

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    The mathematics we use are a human construct, if that's what you're asking. There are many mathematical constructs that are purely pathological (see e.g. "Counterexamples in Topology"-Steen et al and "Counterexamples in Analysis"-Gelbaum et al). But it certainly isn't subjective.
     
  4. Jul 7, 2013 #3
    Personally, I lost my faith in mathematics when I discovered some troublemaker named George Boole made up his own version of arithmetic where 1+1=1. As if that weren't enough, another malcontent named Kurt Friedrich Gödel came along and said you can't prove arithmetic, and by extension, all of mathematics is internally consistent. That means any set of basic rules called axioms can't be proven to be self consistent. If you make deductions from those axioms, there's no guarantee you won't get a contradiction. Everyone just crosses their fingers and hopes. That's not good enough for me. I'm a mathematical atheist.

    EDIT: BTW axioms can't be proven by definition. The Curia just get together from to time to time to write a new axiom, modify an axiom, or replace an axiom. Who do they think they are?
     
    Last edited: Jul 7, 2013
  5. Jul 7, 2013 #4

    micromass

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    Why do you think there is no empirical basis for mathematics? Mathematics is used in physics, biology, engineering, etc. And apparently it leads to correct results. This should be enough for empirical basis.

    If you think that mathematics is based on axioms, then that is wrong. That is only a way to present mathematics to the reader. Real mathematical research doesn't start with "Let's put on some arbitrary axioms and see what we can derive". No math discipline was discovered that way. It usually starts with: "Here is some mathematical or physical phenonemenon, let's investigate this". Then we usually see that what we're doing is very similar to an already existing part of mathematics. This makes us think that we can abstract the mathematics. So we invent axioms to make sense of the abstraction.

    Almost all mathematics that there is eventually has some roots in some physical phenomenon. We don't just make random things up for fun, and then it turns out that it's actually useful. If mathematicians make up something new, then there is usually some reason to do it and some motivation for the concept.
     
  6. Jul 7, 2013 #5
    It bothers me at times. It is one of the only questions that keeps me up at night. I really don't know and I don't think we'll ever know which irritates me. One thing that salvages it for me is that it works. When Maths is applied to real life via physics / networks etc. it makes predictions which are then backed up by empirical evidence. But one could argue that that's the science working and not the maths. So I guess I really have no idea what to think about it and that makes me angry.

    Thanks for making me angry. I hope you're proud of yourself. :D
     
  7. Jul 7, 2013 #6
    Why are you angry? Using the word "faith" with mathematics was what I was satirizing. Suppose I wrote a logic where 1+1=4. Nonsense? Maybe. But biologically, two individuals, one male and one female, will produce two offspring at long term replacement rates and assuming a period of parental survival; 1+1=4. Of course, it isn't exactly how I would describe the model if I wanted to get funded
     
  8. Jul 7, 2013 #7
    I was joking i'm not really angry.

    But I just get irritated that something I enjoy so much and had a lot of well reasoned trust in, cannot be validated 100%. Eg. Godel.


    Edit: Upon a second reading, irritated seems like a strong word. Uncomfortable seems better.
     
  9. Jul 7, 2013 #8
    I don't think anyone really worries about. Even Gödel thought arithmetic was true even if not all true statements in arithmetic can be proven to be true.
     
  10. Jul 7, 2013 #9

    Fredrik

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    Take the field of real numbers for example. It's defined by a set of axioms that was chosen to ensure that real numbers would be useful, in particular that they would agree with our intuitive understanding of space and time. But relativity has taught us that our intuitive understanding of space and time is wrong (in the sense that it's only approximately correct). So I wouldn't say that there's an empirical basis for the real number axioms. I would just say that the real number axioms were motivated by things in the real world.

    Now, does this mean that the real number axioms are wrong? In my opinion, it doesn't. A set of axioms is either consistent or inconsistent, but is neither right nor wrong. The inconsistent sets of axioms can be dismissed as completely useless. The consistent sets should be judged only by how useful they are in applied mathematics.
     
  11. Jul 7, 2013 #10
    Almost all mathematics that there is eventually has some roots in some physical phenomenon. We don't just make random things up for fun, and then it turns out that it's actually useful. If mathematicians make up something new, then there is usually some reason to do it and some motivation for the concept.

    Work by Bernhard Riemann allowed for the development of general relativity 80 years later. Riemann wasn't even a physicist. It sounds like mathematicians are making stuff up (but in a very defined and rigorous manner).
     
  12. Jul 7, 2013 #11

    WannabeNewton

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    Riemann's work was motivated by the classical differential geometry of curves and surfaces developed by Gauss. People don't just arbitrarily come up with mathematical theories.
     
  13. Jul 7, 2013 #12
    But it doesn't change the fact that the math itself doesn't have any application until later.
     
  14. Jul 7, 2013 #13

    WannabeNewton

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    So what? Even in physics we can come up completely pathological constructs. There are known pathological solutions to the Einstein equations that have nothing to do with observed reality. You can even systematically generate solutions to the Einstein equations that have no physical use whatsoever. It isn't only mathematics that has constructions which are out of touch with reality.
     
  15. Jul 7, 2013 #14
    Doesn't that bother you? Isn't there something extremely odd about it. And do you mind defining pathological. I have never heard it used in this context (I study pathological pathways and changes of tissues).
     
  16. Jul 7, 2013 #15

    WannabeNewton

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  17. Jul 7, 2013 #16
    We already use math to describe many of our observations of nature, why can't we expect it to be linked to nature. Isn't the reason that its so good at describing the real world be an indication that it should be linked to nature.
     
  18. Jul 7, 2013 #17

    WannabeNewton

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    I don't see a reason why it should. Clearly we can come up with things that have no relation to observed reality in both mathematics and physics so that right there should tell you otherwise. Just because some of it does doesn't mean all of it has to.
     
  19. Jul 7, 2013 #18
    Argh, your mathematical fictionalist views clash with my nominalistic views. Why can't I picture the root of negative one :cry:
     
  20. Jul 7, 2013 #19

    micromass

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    You can easily picture it as a rotation over 90 degrees. Rotate twice, and you get a reflection = -1.
     
  21. Jul 7, 2013 #20

    micromass

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    Like wbn said, Riemann's work was physically inspired by the study of curves and surfaces. This study was very important that period. For example, the study of the cycloid was very popular and was motivated by physics. There's also spherical trigonometry and study of the earth which is helpful for sea travel.

    These investigations lead Riemann to define things more abstractly and more generally. So what he did is certainly rooted in physical applications, even though his theory might not have had applications itself.

    That is later became useful in relativity is no real surprise. Differential geometry studies spaces which are locally flat. There are many examples of this. It is no surprise that the study of the universe also requires differential geometry, because the universe looks locally flat to us.
     
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