Physics Science

Mark Levi’s book “The Mathematical Mechanic” is a wonderful attempt to integrate physical reasoning with mathematical reasoning. These two strands have historically run in parallel and only occasionally have they been united at least at a pedagogical level. There seems to be a trend among Russian mathematicians particularly in the area of differential equations whereby they use physical reasoning to illuminate the more abstract mathematical approaches that are taken. V I Arnold is an example someone who has been known to integrate the two approaches. Perhaps Levi’s Russian roots explain some of the impetus for this book. As mathematics becomes more and more specialised I fear that fewer mathematicians have the time or even inclination to think about the interconnections between physical reasoning and their own area. Levi’s book is an antidote to that trend and he is to be congratulated for his efforts.

What Levy does is to take a large number of mathematical problems/theorems and show how physical reasoning using concepts such as conservation of energy, torque, resolution of forces, etc can be used to solve what are quite fundamental problems/theorems. In Chapter 2 he uses essentially torque concepts to prove the Pythagorean theorem be a thought experiment involving a right angled prism sitting in a water filled fish tank but attached to a spindle so it can rotate. The fact that it doesn’t (ie there is zero net torque) leads directly to Pythagoras’ Theorem.

Many of the problems turn upon one very basic physical principle and some careful reasoning about how that physical principle applies. For instance in working out why a triangle balances on the point of intersection of the medians the basic idea is a reductionist one and that is to conceptually slice a strip of the triangle. Since this strip balances and all the ones parallel to it will balance one can replicate the same argument for any other side and the point of balance will lie on the intersection of the medians. Levy spends a bit of time on geometrical optics and Fermat’s principle and Snell’s Law and gives a number of physical proofs for various formulas. There is that old favourite of saving a drowning victim by using Fermat’s principle and this is explained in terms of Snell’s law.

An interesting application of the general approach is to prove that the arithmetic mean is greater than the geometric mean by for throwing a switch. This all turns upon the concept of resistance along parallel paths and the result follows very quickly. Levy generalizes that approach to more complex arrangements. He covers Pappus Volume Theorem and applications of Ceva’s Theorem. He also shows how you can compute the integral of sin x by using concepts of potential energy in the context of the movement of the pendulum. He touches on Hamiltonian mechanics and the Euler Lagrange equations and he even provides a hand waving proof of area preservation.

On page 125 there is a table of analogies between mechanics and analysis. For instance zero net work done is interpreted in an analytical sense in terms of preservation of the area. There is an interesting discussion of how an area preservation property can be viewed as a classical mechanical analog of the uncertainty principle in quantum mechanics. If an area preserving map squeezes some region about a point x we gain information about that point however because the map is area preserving it must stretch in the other direction (y) and this means that the range of values in the other direction is large so we lose information in that direction. If we think of the first variable x as signifying position and the second one being y which is identified with momentum, we then have the connection with the uncertainty principle.

I’m not aware of any other books that have systematically brought together this type of physical reasoning and its application to mathematical problems. In bringing together such a wide range of problems Levi has at the very least provided interested people with something to go on with in a more systematic fashion. The beauty of the book is that often a compelling physical reason for a particular mathematical equation can be much easier to remember and can actually illuminate the mathematical proof. One could even contemplate a little subculture of mathematics developing whereby people try to develop more and more inspired physical analogies for various mathematical theorems.

Levy does not assume a great level of mathematical sophistication however readers should have a reasonable grasp of basic concepts such as the resolution of forces, potential energy, kinetic energy and how the can be applied to a problem. There is no heavy-duty calculus or analysis involved and Levy has a very informal and chatty style.

I recommend this book without any reservation – it should have been written many years ago. I think students will find it enriches their understanding of the concepts.
Rating: 5 / 5

I entirely disagree with observations of author in page 3 that mechanics can be approached/developed as a rigorous and pure axiomatic subject and for that matter, to take such exercise for any branch of physics.

David Hilbert proposed in 1920 in his famous 23 problems to axiomatize physics – where it was mentioned as 6th item. This was never accepted by physicists. As a matter of fact, too dependence on axiomatic approach makes one blind, sightless and orthodox.

In 1931, Kurt Godel proved in his remarkable incompleteness theorems that this approach has limitations.

For any scientist, intuition and imagination is fundamentally important to foresee a result. Later, rationale or foundations are provided with aid of mathematics to justify derivations. A debate on supremacy of math or physics is nonsensical as we cannot compare apples with oranges. Essentially, approach in physics is inductive – to look for unification from seemingly unrelated events while we take deductive approach in math based on axioms or a set of accepted premises. Each of this requires a different specialized skill set.

A physicist has to always look for necessary revisions or new theories based on observations keeping his mind and options open. A mathematician has no such liability or problem. Although, scientists generally believe that creative mathematical ideas need to serve purpose of real world – as author recalled his friend saying that mathematics is servant of physics – but there is no such binding on a mathematician who can come up with something abstract.

Much of mathematics was developed without any aid from real world which came to use later; more will be created in future. A mathematician may not require to justify why a particular definition was adopted debating possibilities of alternative set of definitions or make a comparison between those to gain insights into fundamental principles. This kind of analysis has been rarely taken (example, to prove 5th postulate of Euclid). The difficulty with axiomatic approach is that we cannot challenge, go behind premises to understand real nature of things. I’ve heard a professor of mathematics saying that a definition is a definition – we’ve to accept whether it’s useful or not. A physicist cannot take such stance. However, axiomatic approach in mathematics is important as it provides logical foundation or rationale for derivations and over-dependence on intuition sometimes can be misleading – prove to be fatal. We often criticize mathematicians as nothing is acceptable to them without a valid proof even when, a conclusion seems obvious. We rarely understand that, a mathematician has limitations working within the scope of axioms and field of mathematics itself imposes that discipline in the mind of a mathematician. Ideally, one has to strike a balance between both approaches.

From a quick glance, I find author has tried to retrofit that is, found a logic to fit into established results instead of deriving results based on premises. Certainly, he has offered more insights from real world problems into working of mathematics, but conclusions did not flow logically from statements of problems. As for example, in page 6 (section 1.3), he mentioned that potential energy of the first string is AX to drag X from hole A to current position X… this string is supposed to be mass-less and therefore, doesn’t have potential energy and in any case on flat table where string rests, there cannot be any change in potential energy. Actually, this needs to be corrected as “work done” while dragging instead of potential energy and these two are not same concepts. Then, it is mentioned that system is in equilibrium and so, consequently minimal energy (assuming potential energy) and it is difficult to see flow of logic and connection between “work done”, equilibrium and minimality of sum of distances from three corner holes of the table.

I think this is entirely a wrong approach to loosely analyze a situation which is neither intuitive nor rational and if taken by a freshman or new enthusiast in math or physics, will make him profoundly confused. Even for arguments sake, this cannot be considered as a sketch of a rigorous proof, as mentioned by author.

In page 19 (section 2.8), author has proved a 3-dimension version of Pythagorean theorem with thought experiment of a gas filled tetrahedron. The sum of internal forces amounts to zero from first and second laws of motion and it is not clear why the principle of conservation of energy has been brought into play to prove equilibrium and invariance of volume of tetrahedron under translation. A system will remain in mechanical equilibrium if, and only if, sum of net forces (external) and net torques (external again) work out to be zero. In fact, conservation of energy has nothing to do with equilibrium !

I can throw any ball up in the air and while it is in motion (as it goes up and comes down), we can see that there is conservation of energy, but ball is never in equilibrium. With due respect to author, I’ve to say that in this case his line of argument is wrong and this kind of wrong reasoning can lead to paradoxical results !

In proving Pythagorean theorem, author used Pythagorean theorem raising confusion of circular reasoning, mentioned by a reviewer but, actually he proved a surface area version of this theorem from well-known Pythagorean equality for right-angled triangles, which is the basis for deriving magnitude of resultant from three orthogonal vectors. Nevertheless, this can be easily proved by pure geometric arguments.

But, most importantly, if we can deduce geometric results from equilibrium of internal forces, then can we trace similar proof in the reverse path ? Actually, we may assume that 3-dimension version of Pythagorean theorem is available from geometry and then, we can deduce force equilibrium simply multiplying the surfaces of tetrahedron with uniform pressure of gas. Do we need laws of motion to explain equilibrium of tetrahedron then ? We are able to use geometry instead of laws of motion!!! This is absurd ! We need to understand that this confusion is arising out of interpretation of a result rather than derivation of result per se. A naive reader may be tempted to think that Pythagorean theorem is an outcome of physical laws and make correlations (apparent from reviews in this post) but, this is not the case ! For, this theorem is a direct consequence of Euclidean geometry in 2-3 dimension spaces and will hold true regardless of our tetrahedron gas filled or not !!!

However, this wrong interpretation has been justified by author as physical aspect of mathematics in page 4 of his book !

It is too much to say that Physics is Math – Math is Physics, unified… we have not advanced that far yet but, there are possibilities of fundamental truths in the universe as a consequence of mathematical logic.
Rating: 2 / 5

In this unusual book, the author discusses mathematical formulas and theorems using purely physical arguments, thus eliminating the usual detailed mathematical approaches. Some of the mathematical subject areas that are discussed include geometry, conics, integration and complex variables. Some of the physical disciplines that are used are mechanics, electricity, fluid dynamics and statics and optics. I found the level of difficulty to vary throughout the book; much of the material is clear, simple and really quite fascinating, while some of it is rather complex, significantly more challenging and often quite difficult to follow, i.e., real head-scratchers. What didn’t help in the latter category were the several editorial mistakes which became rather annoying in the long run. The writing style is friendly, authoritative and generally clear but undoubtedly assumes a certain level of mathematical sophistication on the part of the reader. In my view, this is a book better suited for careful study at one’s own pace rather than be leisurely read as one would a popular science/math book or a novel. Consequently, serious math/science buffs could certainly enjoy perusing this book and learn a great deal from it; however, it could also be used by math/physics students as a supplementary reference in an advanced math or physics course (as suggested by the author).

As a final note, I disagree with the author’s statement that this book “should appeal to … many people who are not interested in mathematics because they find it dry or boring”. Although I understand (and agree with) the author’s implication that mathematics is very far from being dry and boring, I would expect that most of the people he refers to would have avoided mathematics in their lives and would thus be unwilling to read this book in the first place, or be unable to follow most of the discussions presented if they did try to read it.

Rating: 4 / 5