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Why is Mathematics so successful in Physics?

  1. Jan 24, 2012 #1
    Mathematics plays a central role in Physics, but this doesn't happen in other disciplines. I'd like to see your opinions on these questions:

    1) Why is Mathematics so successful in Physics?
    2) Can that success be expanded to other disciplines?

    1) I think it's because Physics problems have few variables. In classical gravity for example, the variables of gravitational force exerted by a particle in another particle are the masses and a distance. In terms of the differential formulation, the only variable is mass density.
    In classical EM, what causes an electric field is either a charge or a variable magnetic field. And what causes a magnetic field is a variable electric field.
    In mechanics, defining what an acceleration is in terms of position, and with a simple law with mass and acceleration as variables, it's possible to describe all the events.

    I'm not saying it's easy to derive physical laws (without knowing what they are already), these are just examples that show that the fundamentals of classical Physics are only based on a few quantitative variables. And by knowing the fundamentals quantitatively, it's possible to describe physical systems mathematically and to derive solutions to complex problems.

    I think this is the fundamental difference that separates Physics and Chemistry (where this still applies I think) from other disciplines. In Economics, for example, there aren't so few variables in any problem. In Biology and Psychology, among others, I think it's also the lack of quantitative significant variables that doesn't make it possible for Mathematics to play a big role.

    So in essence, I think that there are 2 significant factors that make Mathematics have a central role in Physics: few variables and quantitative variables.

    2) First of all, not every discipline needs Mathematics as being central. In Biology, many problems are solved by observation and don't need quantitative analysis, because many times it's not a quantitative problem. But in Economics, there's plenty of quantitative data, so there are quantitative variables. The problem here is the huge amount of variables... If an economy could be fully described by the GDP, unemployment and savings for example, it would be easy to make a function GDP(Unemployment, Savings). Then it would be "easy" to investigate what causes an economy to be described by this function, and derive the macroeconomic model from the microeconomic model, like in Thermodynamics. But there are too many variables, so this isn't possible... It's also not possible to consider problems in isolation.

    I'd like to see your opinion, especially if you know more about this than I do. Sorry if my arguments aren't very good, I never read anything about this, so I'm just expressing the opinions I formed myself about this.
  2. jcsd
  3. Jan 24, 2012 #2
    Have you read The Unreasonable Effectiveness of Mathematics in the Natural Sciences by Eugene Wigner?
  4. Jan 24, 2012 #3
    No, as I've said I never read anything about this. I'll check that out.
  5. Jan 24, 2012 #4
  6. Jan 24, 2012 #5
    Mathematics is effective in any discipline in which the variables can be measured. I have a former student who is now an economics professor at U of Chicago. His undergrad major was mathematics. He went into economics because the content was interesting inherently, real world stuff, but also to quote him, "because I get to do really cool mathematics". His PhD is on the internet if you'd like to see the mathematics of economics in action.

    More and more areas are now able to use mathematics because of the invention/creation of equipment that allows us to measure things that were never before measurable - like brain waves, electromagnetic responses of the brain to different stimuli. This is a very exciting time to live.
  7. Jan 24, 2012 #6
    It's an essay. I just read it, very useless in my opinion. He keeps calling things a "miracle" or a "gift", instead of analyzing them properly. And his main point is that the effectiveness of mathematics in natural sciences is that it has no rational explanation. Reading this essay has no rational explanation too :biggrin:

    I agree. But effectively formulating a theory mathematically seems restricted to Physics.
  8. Jan 24, 2012 #7
    It's been successful longer in physics simply because a significant number of the variables in physics are easy to measure. Think the length and mass of a rod compared to the amount of chlorophyll in a plant leaf and the amount of CO2 it uses. The first has been measured for centuries. The second - not very long.
  9. Jan 24, 2012 #8
    Because math is the study of formal patterns and nature is imbued with formal patterns.
  10. Jan 24, 2012 #9
    That statement is completely wrong. Even if you are doing biology your results must be expressed by a set of numbers if you are actually doing science.There is no acceptable way in which you can determine the validity of your results except mathematics.So my opinion is that there is no science that does not require maths.
    I don't know what you mean by "effectively formulating a theory" but if you are a scientist you must give an estimate of the validity of your result using mathematics.If you are speaking about theoretical science this is still valid because you must quantize your predictions in some kind of meaningful way.
    Last edited: Jan 24, 2012
  11. Jan 24, 2012 #10
    In biology you use of a lot of differential equations in studying predator-prey dynamics and distinctly remember reading many mathematical inclined papers in my senior year while taking evolutionary ecology. Many of the research papers we read had partial differential equations, matrix algebra, and so forth.

    The entire realm of physical chemistry, theoretical chemistry, computational chemistry, inorganic chemistry, quantum chemistry, metallic inorganic chemistry and so forth utilizes a tremendous amount of mathematics.

    Mathematics is tied and has been very successful in so many various fields in science, other than physics, that I don't even know any fields where it isn't utilized.
  12. Jan 24, 2012 #11


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    You can say that the water is:

    hot, cold, or lukewarm
    this gives you three bins to throw each sample of water into. Yay.

    If you want to go further, you could be like:
    very hot, hot, lukewarm, cold, very cold.
    FIVE bins now! That's exciting! Especially if different things happen at cold than happen at very cold; now you can start making laws: "at very cold temperatures, water freezes... at cold temperatures, it's uncomfortable to go out without clothes, at very hot temperatures, water boils"

    But then people will start to notice that there's some wiggle room with "very hot" so they say, "well, let's have more very hots:"

    very hot
    very very hot
    very very very hot
    very very very very hot
    very very very very very hot

    Now you can simplify the representation of all this with a number system... and start defining your laws better about what happens at each stage of hotness. This is essentially approaching quantification from qualification.

    Then you find that particular aspects of quantification are consistent, independent of what you're qualitatively studying (i.e. 1 and 1 always add up to 2). And you have there the field of mathematics. Because of the high degree of states and outcomes available in the world, this higher order description of things becomes necessary to explain a myriad of changes in the physical world.

    So basically, mathematics is just a more accurate language (describing each degree of temperature rather than binning it into "hot" or "cold").
  13. Jan 25, 2012 #12


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    This is incorrect, no matter what you are doing in biology if you have data you are going to need to perform statistical analysis. If you're dealing with any kind of biochemistry you are going to calculate things like molar concentrations, in population genetics you'll need to calculate allele frequencies even simple experiments like total DNA assays will require you to feed the data into an equation to get the results.

    Sure the average biologist might use less mathematics in their daily research than a physicist but mathematics is an integral part of all human affairs.
  14. Jan 25, 2012 #13
    For example the evolution theory by Darwin didn't use mathematics as far as I know. It mainly analyzes qualitative variables such as physical characteristics of animals. Sure some are quantifiable, such as height, etc, but that doesn't make mathematics have the main role in that theory. It's not used to formulate the theory, just to give some support on quantifiable variables.
    And for example Faraday's work - he didn't use mathematics, but he made very important work on electromagnetism. Wouldn't you call his work science, just because he didn't use mathematics?

    My point is: Mathematics is only used to formulate theories in Physics (I don't consider a mathematical model for very restricted phenomena a theory, eg. modelling CO2 absorption by plants). The other sciences are developed mainly by qualitative analysis, not by using mathematical equations.
    Last edited: Jan 25, 2012
  15. Jan 25, 2012 #14
    The reason why Mathematics is so effective in Physics is because the Mathematics that is used by physicists was developed for the purposes of using it in physical research. It almost sounds like a circular argument, but it is true.
  16. Jan 25, 2012 #15
    This is an interesting article which walks through the different schools in philosophy of mathematics.

  17. Jan 25, 2012 #16


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    This premise is false. One can find numerous examples, e.g., describing the EM losses from magetically confined plasmas that have many variables, e.g., position, velocity, acceleration, nuclei (by each species) and electron density, charge density, magnetic field density, . . . , and these variables are functions of time, possibly over many magnitudes of scale. This is one example of a highly coupled system of nonlinear partial differential equations. One can find many examples in applied physics or engineering.

    Mathematics is central to engineering, economics, thermodynamics, computational fluid dynamics, structural analysis, computational material science, computational chemistry, . . . .

    Obviously, any quantitative science requires math!

    However, one can certainly make qualitative statements in all of the above fields, e.g., heat flows from hot to cold, or stress causes distortion of a solid, and permanent deformation if the stress exceeds the yield strength, . . . . .
  18. Jan 25, 2012 #17
    In my opinion mathematics is basically a smart way of counting things. Anything that needs some quantity to measure needs to count them and thus needs mathematics.
  19. Jan 25, 2012 #18


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    That is a large part of mathematics, but mathematics is a language.

    It not only deals with quantity, but it deals with things like change and general forms of representation to name a few.

    As of the last few centuries, it has exploded into so many areas that counting has become a very minor part of what mathematics is today.
  20. Jan 25, 2012 #19
    Mathematics is used extensively in http://pcp.lanl.gov/MATHMPG.html [Broken]. Probably the first thing thrown at you when you begin learning about the topic is the differential equation modelling logistic growth in populations.

    And you are forgetting Computational Biology and a lot of biochemistry.
    Last edited by a moderator: May 5, 2017
  21. Jan 25, 2012 #20
    Yes, but we can only analyze those complex systems with a lot of variables because we know the fundamentals, that are understood from simpler problems with few variables (I said that latter in the post, just not so explicitly). If the only system we could analyze was that you suggested, it would be very difficult to find mechanics and EM laws. So let me reformulate... There are 2 things about these variables that I think make mathematics assume a central role in Physics: quantitative variables and the ability to study systems with few variables.
  22. Jan 25, 2012 #21
    Biology USED TO BE qualitative but no longer. Every aspect of biology has quantitative measurements today.

    The number of variables you measure and analyze can be controlled in any science. This is the reason "lab rats" are used. They are bred to have identical characteristics - all "twins", clones, so changes can be made and measurements can be done to determine the causes of disease, the effectiveness of medications and so on.

    Psychology is now highly analytical and quantitative. Again the experiments are controlled. Variables are limited and changes are measured in everything from brain chemical concentrations to electrical impulses. Since Physics is the foundations science and mathematics has always been the language of physics and all sciences can be studied in terms of the foundation science, mathematics is equally valuable and used in all of them.

    This is the reason students in the lower grades need to be informed very soon that if they have an interest in any experimental field they MUST study mathematics until they are comfortable using it. I can't tell you how many students I've known who have gone into things like medicine and biological sciences and who came back saying "you were right - there is mathematics in everything I'm studying". Of course one must learn the language of the science in addition to mathematics. You must know what torque is intuitively as well as just a formula for it, what the names of the bones of the body are, the names of aberrant behaviors in psychology and so on. But mathematics is how we actually develop definitive measurable and reproducible results in all the sciences.

    So again - classical physics dealt with things that were easily measured and obviously related so mathematics was used in it from the beginning. Biology didn't. It does now. As does chemistry, psychology, sociology and so on.

    Think about astronomy. It began as fairly descriptive but certainly isn't now. Why? Because we now have the ability to measure all sorts of things in astronomy even though they are very very far away.

    So again, you have to be able to measure things and develop precise relations between them before you can use mathematics. Physics was the easiest. Measurement became more sophisticated. So now mathematics is applied to pretty much everything. The LOVE chemical in the human brain has even been discovered and I'm sure will be measured and where and how it is produced in the body will be studied and formulas will be developed to relate it to other phenomena.

    Today nothing is safe from a mathematical analysis.
  23. Jan 26, 2012 #22
    Its not correct. Counting is all about mathematics. Whatever higher level mathematics you do all of them are actually counting something (yet not doing it in the same way as counting all numbers from 0 to x then to dx or whatever. We have invented smart ways of doing exactly the same thing.)

    The only thing that may be distinguished from mathematics yet often can be confused with mathematics is logic.
  24. Jan 26, 2012 #23
    Exactly. Abstract mathematics is a generalization of things originally measured and related in a quantitative manner. Mathematics allows to to "go where no man (or woman) has gone before" but it originated in reality.
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