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### The Role of Mathematics and Rational Arguments in Science

Mathematics is a tool used by scientists to help them construct models of how the universe works and make precise predictions that can be tested against observation. That is really all there is to it, but I had better add some more or this will be a really short essay.

For an activity to be science, it is neither necessary, nor sufficient, for it to involve math. Astrology uses very precise mathematics to calculate the planetary positions, but that does not make it science any more than using a hammer makes one a carpenter (Ouch, my finger!). Similarly, not using math does not necessarily mean one is not doing science any more than not using a hammer means one is not a carpenter. Carl Linnaeus’s (1707 – 1778) classification of living things and Charles Darwin’s (1809 – 1882) work on evolution are prime examples of science being done with minimal mathematics (and yes, they are science). The ancient Greek philosophers, either Plato or Aristotle, would have considered the use of math in describing observations as strange and perhaps even pathological. Following their lead, Galileo was criticized for using math to describe motion. Yet since his time, the development of physics, in particular, has been joined at the hip to mathematics.

The foundation of mathematics itself is a whole different can of worms. Is it simply a tautology, with symbols manipulated according to well defined rules? Or is it synthetic a priori information? Is 2+2=4 a profound statement about the universe or simply the definition of 4? Bertrand Russell (1872 – 1970) argued the latter and then showed 3+1=4. Are the mathematical theorems invented or discovered? There are ongoing arguments on the topic, but who knows? I certainly don’t. Fortunately, it does not matter for our purposes. All we need to know about mathematics, from the point of view of science, is that it helps us make more precise predictions. It works, so we use it. That’s all.

I could end this essay here, but it is still quite short. Luckily, there is more. Mathematics is so entwined with parts of science that is has become its de facto language. That is certainly true of physics where the mathematics is an integral part of our thinking. When two physicists discuss, the equations fly. This is still using mathematics as a tool, but a tool that is fully integrated in to the process of science. This has a serious downside. People who do not have a strong background in mathematics are to some extent alienated from science. They can have, at best, a superficial understanding of it from studying the translation of the mathematics into common language. Something is always lost in a translation. In translating topics like quantum mechanics—or indeed most of modern particle physics—that loss is large; hence nonsense like the “God Particle”. There is no “God Particle” in the mathematics, only some elegant equations and, really, considering their importance, quite simple equations.  One hears question like: How do you really understand quantum mechanics? The answer is clear, study the mathematics. That is where the real meat of the topic and where the understanding is—not in some dreamed up metaphysics-like the many worlds interpretation.

Closely related to mathematics are logical and rational arguments. Logic may or may not give rise to mathematics, but for science, all we require from logic is that it be useful. Rational arguments are a different story. Like mathematics, they are useful only to the extent they help us make better predictions. But that is where the resemblance stops. Rational arguments masquerade as logic, but often become rationalizations: seductive, but specious.  Unlike mathematics, rational arguments are not sufficiently constrained by their rules to be 100% reliable. Indeed, one can say that the prime problem with much of philosophy is the unreliability of seemingly rational arguments. Philosophers using supposedly rational arguments come to wildly different conclusions: compare Plato, Descartes, Hume, and Kant. This is perhaps the main difference between science and philosophy: philosophers trust rational arguments, while scientists insist they be very tightly constrained by observation; hence the success of science.

In science, we start with an idea and develop it using rational arguments and mathematics. We check it with our colleagues and convince ourselves using entirely rational arguments that it must be correct, absolutely, 100%. Then the experiment is performed. Damn—another beautiful theory slain by an ugly fact. Philosophy is like science, but without the experiment[1]. Perhaps the real definition of a rational argument, as compared to a rationalization, is one that produces results that agree with observations. Mathematics, logic, and rational arguments are just a means to an end, producing models that allow us to make precise predictions. And in the end, it is only the success of the predictions that count.

Additional posts in this series will appear most Friday afternoons at 3:30 pm Vancouver time. To receive a reminder follow me on Twitter: @musquod.

[1] I believe this observation comes from one of the Huxelys but I cannot find the reference.

### 11 Responses to “The Role of Mathematics and Rational Arguments in Science”

1. Bent Rothenberg says:

Equations are purely math,in nature time contnues,to me we should use => and not =,this implies that,once decided what is the beging of our function f(x)=y combined with the time t the diffence between math and physics would be math f(x)t=yt physics f(x)t=> yt,it means no going backwards,so to create the Big Bang,nesseraly there must have been something before, the happening,it could not have come out of nothing.
Look at a cell under electromicroscope and look at our galaxie,the look very much alike,the style of making organic living things,has used the forms allready created in the universe.Our solar system looks like an atom or rather a mixture of H = He²,where Jupiter would bee the electron of H nad Saturn Uranus the elctrons of He²,the smaller planets would bee quarks and other exotic particles!

2. Dean says:

As a Mathematian I find science (especially quantum mechanics) to be an incredibly beautiful applications of mathematics as a language.

That said from math looking into physics, I often consider myself to be exploring the language in which the universe speaks to us; hether we understand what it is saying or not. There are many areas where Mathematics doesn’t seem to have a completely real application but I normally consider that to be akin to studying language structures which we have not experienced yet. That something can be phrased mathematically doesn’t mean it is true, but means that it represents a rational argument (though not necessarily scientific argument, string theory is an example). When physics searches for a grand unified theory I can only hope that at some point in time the more abstract theories of Mathematics can be joined real observations, describing real systems.

3. Mike Will says:

Did mathematics exist before mathematicians? If so, then is the ‘language of the universe’ deductive logic? Hmmm. I look at the recent success of inductive methods (algorithms instead of formulae) and wonder. Byron, my take away from the essay is the first sentence and last sentence. Thanks again.

4. Cibele says:

Math is definitely the language of the Universe. Congratulations!!!

5. [...] For an activity to be science, it is neither necessary, nor sufficient, for it to involve math. Astrology uses very precise mathematics to calculate the planetary positions, but that does not make it science any more than using a hammer makes one a carpenter (Ouch, my finger!). Similarly, not using math does not necessarily mean one is not doing science any more than not using a hammer means one is not a carpenter. Quantum Diaries [...]

6. Cormac says:

I love the way reading an equation backwards or sideways often giwes a new insight into the relation between the variables, especially in physics. F = ma defines force as an influence that causes a change in motion while a = F/m defines mass as a resistance to motion etc.
People who dislike math miss out on the power of equations to force us to consider the relation between variables from every angle

7. Uncle Al says:

Organic synthesis proceeds tolerably well with little beyond arithmetic. Bullvalene is remarkable not only for having no static structure (pundits stated doubts prior to synthesis). The two-step synthesis, with no added reagents, ends with the byproduct being bullvalene’s crystallization solvent.

Math has it uses, e.g., bosinos and sleptons. Imagination and its reduction to practice can be louder in context.

8. fluidic says:

Physics and the physical universe was created and preceeded math. Math is the best tool that was naturally developed to take good care of the measuring, slicing, counting, adding, subtracting, multiplying, dividing, predicting, scaling, looking, viewing, proportioning, etc our physical objects in the universe following the BGBG and creation.

Why would anyone require math if there is no use for it? since after BGBG, the universe needed math and we are still inherting math just like students inherit MATHLAB, CATIA, FLUENT etc to solve their difficult problems. They keep contributing to these software giant developers like we contribute to the development of math tools everyday.

Physics preceded math which preceded all sciences.

9. chrisanto says:

re: Bertrand Russell’s ‘Principia Mathematica’ proof of 1+1=2 (or 2+2=4) based on logical propositions….I understand that the entire PM relies on 3 base axioms which must be true for all of mathematics to follow logically. I find it fascinating to consider discovering a case where these axioms fail.

10. Nir says:

I agree with most of what’s written, but not all. You yourself describe science as being about making precise predictions that are verifiable (or more accurately, falsifiable). Yet you then include Carl Linnaeus’ work as science. While it is important to build a scientific vocabulary, what exactly are the precise predictions of his theory, tested against experiment? Observing things and writing them down does not constitute science, and this is the objection that I (and many other physicists) have to so many areas of other sciences. Darwin’s theory highlights the difficulty of verifying theories without mathematics: deprived of mathematics and numerical agreement, one is generally reduced to much more general observations that are very easily observed by a variety of theories. When Darwin’s theory was introduced, there really was not any competition for it (I don’t count creationism as competition, because it’s not even attempting to be a scientific theory). Without any understanding of genetics (not known in Darwin’s time), it is virtually impossible to have any reasonable model of evolution. Every time I’ve ever read the key paragraphs of Darwin’s theory, I’ve always thought to myself: this is practically tautological, minus the stroke of genius in understanding inheritance of traits.
This is why theories need math. The theory of evolution currently says something very much: populations become more fit, where fitness is defined as reproductive capacity, because segments that reproduce more dominate the next generation and give their children the same advantages. To people who grew up without believing in Creationism, this statement just isn’t a big deal. However, quantifying the effects of randomness versus directedness in evolution, i.e. drift vs niche evolution, is a quantifiable aspect of the theory that has sparked much debate in the last couple of decades. And making a statement about which of these forces is dominant, and by how much, is a quantifiable, testable prediction, that of course, requires math.

11. Md Santo says:

Decreasing Role of Math and Rational Arguments in Science :

• Within Data (D), Information (I), Knowledge (K), Wisdom (W) or DIKW continuum which was becoming human scientific mindset model since 17th century, the role of math and rational thinking is very crucial in Science. It is represented in step-by-step “deducto – hypothetico – verificative” way of scientific thinking

• But, beginning the dawn of 21st century, where the continuum of DI are very matured and advance, further step is KW era in which the role of Knowledge becoming very prominent. In this situation, the border of Science shifted to the right where the border of Science with Pseudo Science or Metaphysic becomming blurred.

• The important changes is the behavior of DI as object without consciousness becoming KW acting as subject with consciousness and freewill. Science within domain of DI as “Scientific Knowledge” becoming “Knowledgeable Science” within domain of KW with becoming less crucial of the role of Math and Rational arguments

• How do we addressing the era or domain of post DI or within domain of KW and beyond? In this situation for the sake of future Science we should embark from DIKW model with paradigm of ““Mind Brain or Human Being is the source and center of Consciousness” toward the paradigm of “The Universe or the Nature Knowledge is the source and center of Consciousness”

• To clarify the above statements, should you visit the following URLs :
http://bit.ly/GARHO2 (“Important considerations why the limit of Science should shifted to the right” ),
http://bit.ly/H755iW (“2012 is “Jump Time” to make “Great Turning” from DIKW continuum toward Nature Knowledge continuum within Science evolution”) and http://bit.ly/I9pfUp (“Comprehensive Guide to Future Science Environment : Decomposing Knowledge with Inverted Paradigm Method”)