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Andrew Zimmerman Jones

Mathematics and Physics

By April 4, 2011

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The language of physics is mathematics. In fact, mathematics is the language of science as a whole, as I reiterated recently in an engaging exchange with one of our readers. (Another part of the exchange is outlined in the article "Can Science Prove Anything?")

As mentioned in that earlier article, this reader was concerned about how I instructed people to "analyze the data" in my Introduction to Scientific Method article, when I said:

use proper mathematical analysis to see if the results of the experiment support or refute the hypothesis.

His concern is that mathematics can be misleading, used to obfuscate results. In fact, he was concerned that scientists apply "arbitrary constants" to make the equations work out the way they want to. After all, he suggested, wasn't this exactly what Albert Einstein did when he proposed his cosmological constant?

Well, let's see about that ...

The role of mathematics in physics

Let's begin with the note that I said "use proper mathematical analysis." Certainly, there are many types of mathematical analysis which are not proper for a given set of data, and I am not suggesting using those. Assigning arbitrary constants would fall in line, by my way of thinking, as an improper mathematical analysis.

The reason mathematics is the language of choice for science is twofold.

First, mathematics allows for a great deal of precision. This precision isn't just in the use of a specific number to a given degree of precision, but also in the way the formal structure of the language allows those numbers to interact in very specific ways. It should always be the goal of a scientist to make their predictions and analyze their data with as much precision as possible.

Second, the mathematical structure is one of the best ways to discern patterns within the data, which non-mathematical analyses would often overlook. This isn't to say that there aren't occasionally some benefits to a non-mathematical approach, but in aggregate they are far outweighed by the benefits gained through mathematical analysis.

Arbitrary Constants

The concern that physicists assign arbitrary constants also deserves some greater investigation. At first, I dismissed this, because physicists actually take a great deal of care to conduct experiments to measure the value of their constants to as much precision as possible, especially for the fundamental physical constants.

However, there is a point to be made here. In the realm of theoretical physics, there are some people who make all kind of theoretical predictions, and they can do so because the values and properties of certain portions of physics equations are not yet really well-defined, because we can't yet conduct experiments that would allow us to nail down those properties. Physicists can really make a large number of fairly outlandish claims and even come up with the mathematics to support it. You can create models of inflationary universes, oscillating universes, multiple universes, fractal universes, and so on ... all by making different assumptions about the mathematics involved in describing our universe.

In fact, the case of Einstein's cosmological constant is classic demonstration of a scientist going awry in a highly mathematical theory ... or is it?

Einstein's Big Blunder: Cosmological Constant

Shortly after developing his theory of general relativity, Einstein was informed of a problem. It turned out that his theory resulted in a universe that was inherently unstable and would either expand or collapse, based upon the curvature of spacetime.

The problem is that Einstein believed (along with every other physicist and astronomer of the day) that the universe was essentially static and did not, overall, change over time. It certainly didn't expand or collapse.

In order to fix this, Einstein introduced an extra term into his equation which represented, in the words of Brian Green (in his new book The Hidden Reality), "the amount of energy stitched into the very fabric of space itself." If this value was exactly right, it would result in the static universe that he and everyone else believed existed, based on the known evidence of the time.

Years later, when the expansion of space was discovered, Einstein called this the biggest blunder of his life.

However, is this really a case where the mathematics can be seen as the failure? I don't think so.

Think about it: Einstein's mathematical theory of general relativity is the very theory that predicted the universe should be expanding. Until that point, no one had given any thought to this possibility.

Note: Some had, since the time of Sir Isaac Newton, wondered why the universe didn't collapse under gravity's attraction, but that's another situation.

The failure was not a failure of mathematics, because the mathematical prediction yielded a new insight about the changing nature of the universe which, ultimately, turned out to be true. The failure was that Einstein did not trust his mathematics enough to even investigate this possibility, but instead sought a way to negate it. Why? Because he had no evidence to support such a prediction.

In this case, the mathematics was correct while the evidence was incomplete and misleading!

So, instead of proving the case that a proper mathematical analysis isn't part of the scientific method, the case of Einstein's cosmological constant is a perfect example of mathematics being essential to scientific investigation ... when applied correctly.

In fact, Einstein was even less off base than he knew, because it appears that the universe does have a certain amount of energy stitched into its very fabric. The cosmological constant is still in play even today ... in the form of dark energy, one of the newest and most intriguing mysteries of physics!

Comments

April 4, 2011 at 9:28 am
(1) Charles Ivie says:

Mathematics is a tool, a very powerful tool. But like any tool it can be miss used. As it turned out Einstein’s blunder was actually a window into a deeper reality. It only seemed like a blunder because it revealed what turned out to be an incorrect assumption about the nature of the universe. As a scientist I recognise that my appreciation of the”real” world is actually a collection of models that allow me to interpret the messages from my senses deliver to my mind. The most powerful element of this model is mathematics. In Einstein’s “blunder” it was mathematics that revealed what we now recognise as the truth.

April 11, 2011 at 9:49 am
(2) Deto says:

More than 1400 yrs ago someone said that the universe is expanding that was God him self in the holy Quran:”
We have constructed the heaven with might, and verily, it is We who are steadily expanding it…. ” (Surah 51:47). …
Probably Einstein proved it and that only increases our belief in Great God.

April 11, 2011 at 11:05 am
(3) sky says:

Deto i think you are right but we can’t go for only statements,we have to prove them and also apply for our daily livelihood these all are our scientists did.

god is unique. He gave power to think and he gave clues to us for live life happily

April 11, 2011 at 12:18 pm
(4) pappu kumar says:

sir,u r absolutely right ..we can not define physics without mathematics……

April 13, 2011 at 8:20 am
(5) Barend J.\alberts says:

The throne is reserved for mathematics to be the language of physics, but for it to become an effective one, science must first acquire a sound foundation, explained in such a way that “a child can understand it”

Sound foundation refers to a full understanding of the various forces and fields like gravity, magnetism, weak and strong nuclear forces, Coulomb’s fields, EM waves, protons, etc. There are five (5) questions to be answered in full, i.e. what does each field/force consists of; what is it produced from; how; why; and with what?

For example, if a photon has no rest mass, it does not exist in real terms, obviously, but this “nothing” can become “something” travelling at the speed of light. To the five questions mentioned above must therefor be added:

a. Does a photon actually travel like a cannon ball?
b. And/or, does it travel like an ocean wave, where the wave moves forward, but it’s particles does not
c. Does a photon stay the same size, i.e. one all the way from the sun? If not, the photoelectric effect could not be

Barend J. Alberts
37 St Claire Close
Saldanha 7395
South Africa
+27227142502

April 15, 2011 at 5:59 pm
(6) Ken Koskinen says:

When Einstein introduced his constant the Milky Way was incorrectly thought to have been the whole universe. Astronomers assured him the stars were basically fixed. It was only after a visit with Edwin Hubble that he was informed of the expansion.

However as he came up with his cosmological constant nothing was measured via any experiment. Such a thing was impossible in his day. He used the mathematics of General Relativity to find the value needed in order to tame his equation.

Einstein’s biggest blunder was he could have been first to predict the universe was expanding; even before the astronomical observations!

In recent times physicists resurrected the cosmological constant but in light of the dark energy detections. The constant is now used differently than Einstein had. However, caution is advisable since we don’t yet know the cause of dark energy. It could be due to something else. If it’s due to some unknown particle/field then it could belong in a category called “quintessence.” If the later is the case (or due to some other cause) then dark energy might not be universally constant but could vary somewhat due to differing things/conditions in localities in space/time. We just don’t know yet; but the scientific process proceeds. Who knows? We might have to delete the cosmological constant or alter it again.

In any case mathematics per sec doesn’t really error but people do. Our verbal and mathematical theories can be wrong … okay when that happens … salvage the good pieces and give it another shot! http://antspub.com

April 16, 2011 at 10:18 am
(7) Ken Koskinen says:

Deto, are you reading too much into that verse?

We have constructed the heaven with might, and verily, it is We who are steadily expanding it…. ” (Surah 51:47). …

Who are the “we” who constructed the heaven & who are steadily expanding it?

The universe isn’t a heaven or the assumed abode of Allah or any other gods. If “heaven’ refers to the sky & space above Earth; it hasn’t been expanding and this is what early non-scientific peoples saw with the naked eye. Gravity holds our galaxy together and of course our planet & solar system is within it.

The scientific theory of expansion doesn’t characterize our universe as having been “steadily expanding.” If inflationary theory is correct it started with a tremendous rapidity. This was followed by a slower pace but then about 5 billion years ago the pace again increased via dark energy.

It’s fine to have a religious belief but this verse doesn’t describe the modern theory of expansion of our universe.

April 17, 2011 at 11:14 am
(8) Ken Koskinen says:

Barend J.\alberts

Mathematics is already an effective language of science but so is verbal language. Both are already enthroned. Science has a sound foundation; look at the computer you use! Einstein’s appeal goes to using as much simplistic verbal/visual communication as possible. Children can’t do the math!

Science constantly proceeds towards fuller understandings. Your 5 questions are fine but you repeated the question of “what.” There are many other questions.

(a) Photons are not like cannon balls. (b) light waves are not like ocean waves; they break up on beaches &/or on other obstacles (c) the photoelectric effect is sensitive to the energy of light/photons, not to so-called (undetected) size.

http://antspub.com Download my free “The 3 S’s of Science & the Physics of Humpty Dumpty.” I hope it helps.

April 19, 2011 at 10:00 am
(9) Barend J. Alberts says:

You can drive a car without knowing how the engine works. Or fall off a chair, launch a satellite with amazing precision without knowing how gravity is generated. But one cannot resort to mathematics unless one understands the fundamental, basic forces and fields of nature, such as gravity, strong and weak nuclear, magnetism, Coulomb’s fields, etc.

Web site http://www.benalberts22.com explains those basics

It is because of that lack of understanding that scientists resort to preposterous and ridiculous concepts like string theory, Big Bang, strong and weak nuclear forces, quantum electrodynamics, particle accelerators, EM, photons, and hundreds of others

Barend J. Alberts

October 31, 2012 at 2:50 pm
(10) MIKE KEULER says:

You do not need or can anyone explain completely all of the forces that keep things in order or make us question their very existence. God created and maintains his universe, (we call it), the heavens, i.e., Isostasy, and such. He has allowed very gifted creations such as Einstein to understand many things, but He will not allow us to understand all of it. God’s thoughts are compared to ours, high as the heavens as ours are to the earth. A mind boggling difference, although we keep trying.

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