[Originally posted on July 11, 2012. Updated considerably on July 12, 2012]
“Heavy use of equations impedes communication among biologists.”
That’s the title of a recent paper in PNAS that Prof. Abinandanan recently highlighted on the blog nanopolitan [^]. (One of the blogging tags/labels he used for that post was: “Publish/Perish”!) Let me cite the same passage that he does:
…The density of equations in an article has a significant negative impact on citation rates, with papers receiving 28% fewer citations overall for each additional equation per page in the main text. …
In the comment I made at that post, I recommended a book that is just so excellent. I wish I had access to a book like that during my XI and XII, even during the first year of engineering. Let me more or less just copy-paste what I said in my comment at that blog. Here we go:
By way of some help to our biologist friends (and sometimes also to our engineer, physicist, and even mathematician friends(!)), I would single out one book over all the others that I have seen:
E. Batschelet, “Introduction to Mathematics for Life Scientists”
Preview it on Google, here [^].
The side-links from the Google Books Preview page going out to the Indian book sellers’ sites do not lead to this bit, but at least until recent times, in India, there used to be a very inexpensive Narosa edition (of the 2/e of the book) available at Rs. 195; its ISBN: 3-540-78012-2.
On Amazon, I remember having seen this book receive just a single customer review, an admiring one with a 5-stars rating. I recently checked out the page once again, and found that two more customer reviews have appeared, both with 5-stars ratings [^]. The only reason I don’t wish to add my own review at Amazon is: laziness! Else, you would have a fourth 5-stars review.
For those who are a bit more ambitious about mathematics, the books on history of mathematics by Prof. Maurice Kline should continue to be a particularly enlightening source.
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That is what I effectively said at nanopolitan. Now, let me add a few more links to recent blog posts. These are part 3 [^] and 4 [^] of the series on “The Mathematics of Biodiversity” by Prof. John Baez. I am still going through his series (discovered this blog of his only today), but I already liked the directness with which he states in part 3 the physical correspondents (though not the historical roots) of the Fokker-Planck equation, with as brief an explanation as possible, and with a truly illustrative diagram to accompany it. Delectable!
For an example of how to write a pageful and still not let the reader get the real clue about the equation’s meaning, refer to the Wiki page[^]. Read the Wiki page completely, without skipping a single word or mathematical term. If necessary, read aloud to make sure you did that. Then, take a moment or two to gather what you might have got “to take home” from that reading. Then, once again go back to Prof. Baez’s informally made statement about it. Think. What made the difference? Answer: The fact that Prof. Baez here happened to state the physical meaning of the mathematics. That’s all.
… And, that, BTW, is the hallmark of any neat explanation or exposition of mathematics. Tell people the physical roots of a mathematical idea, and you have essentially conveyed that idea to them. … Rigor can always wait, it can always come later on.
Too many mathematicians, physicists, and even engineers, commit that mistake, however. They begin without spelling out the physical context of mathematical ideas, which leaves the reader/student blindly groping for it in the dark, so to speak, and then, to make the matters worse, these mathematicians (etc.) commit two further sins:
(i) thrashing the hierarchy by jumping into topics in any random sequence, refusing to go from the basics or fundamentals to the derived or advanced, from the simple and elementary to the complex and composite; this practice introduces obstacles on which the student must stumble even more during that blind groping of his, and then,
(ii) introducing very high level of rigor right from the word go—thereby making such high unrealistic demands on the reader’s/student’s mind that it gets stunted by the sheer abstractness of the symbols, with the end result that he shrinks back in the fear of the mathematician’s (or physicist’s or even engineer’s) supposed genius. That last, BTW, could easily be their motivation.
What people fail to realize, time and again, is that mathematics can still remain mathematics even if it doesn’t carry as much rigor as is possible in the most formal presentation of it. Presentation and formulation are two different things. And, among the two, it’s the formulation (or the act of conceiving or creating an idea) that is basically related to understanding. Understanding requires reaching exactly the same mental viewpoint as was reached during creating an idea.
Rigor only makes the already developed understanding more precise; it does not create it in the first place. The use of mathematical symbols serves only to condense the presentation—provided you already know what ideas/concepts of what meaning are going to be used in that presentation, and, further: how the flow involving these ideas is going to occur in that presentation. If you don’t know all those three elements beforehand, symbols, by themselves, only serve to reduce you to the level of a primitive tribal facing a mystic ritual being performed by his priest. Symbols, sans meaning active in a mind, serve only to stunt that mind, whether in a mystical context or a mathematical one.
It has always struck me how wonderful it is to read the original writings of those 19th century physicists. For example, go through Maxwell’s original derivations of those EM equations, and then, contrast that piece of writing to a typical modern, post mid-20th century, treatment of the same (e.g., Feynman, let alone Jackson). The difference is: delineation of the physical roots of ideas vs. its absence for the most part. In Feynman’s case, it is not just an absence; it is: seductively presented arguments against the very idea of having to look for any physical roots. Feynman was a bit rationalistic, not just in his understanding of what a physical theory must be like but also in his presentation of it. Little wonder he ended up having in his theory particles that travel back in time, “arriving” in the here and now from the future that is yet to happen.
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BTW, if you have been reading this blog for some time, you might remember that I had expressed some issues that I had concerning the position about the relation between mathematics and physics, which was taken by David Harriman in his book on induction; see that post of mine, here [^]. I don’t know all the sources of this position of his (nor am I terribly interested in knowing about it). However, given his enormous influence and the charming style of presentation, it could easily have been Feynman (or, if not him, at least the same sources as influencing Feynman). What I now say (and point out or link to) in this post only augments what I had said back then.
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And, coming to the recent blog posts/comments concerning the nature of mathematics and physics, there are a few comments that someone else made that I found just so wonderful.
Dr. James Putnam, while commenting on one of Prof. Scott Aaronson’s blog posts, neatly pins down how mathematics can indeed become arbitrary once severed from its physical roots.
Though there are far too many comments on that thread, and a lot of them are quite interesting in their own right for many different reasons (!), as far as this issue of the nature of relation of mathematics and physics is concerned, the relevant comments by Dr. James Putnam are: # 244 [^] and # 269 [^]. Here are the relevant excerpts:
…The model dictates the meaning and correctness of the math. It is the model to which the values and their variations pertain. It is the model that establishes the properties that are represented by the math.
… I think the debate must be about the physics. If the physics is wrong then the math doesn’t need to be evaluated. If the physics is valid then the math is valid. …
…The mathematics helps us to keep our thoughts straight. It is the orderly processing of our thoughts about relationships between properties. In the case of physics, it is the relationship of physical properties. The math does not tell me what those properties are. I tell the math those conditions….
and, esp., this one!:
…It then even becomes possible for me to add electric charge to thermodynamic entropy. …
To see how, the comment in original. And, also this, from the same comment:
… I see ‘engaging in deep ruminations’ about physics as being the first requirement to be met before establishing the suitable mathematics for representing those deep ruminations. …
Do see these comments in their context, by visiting the original thread. (And, BTW, note, as far as the main issue of that thread goes, there then was a follow-up post, with a lot of comments to that post too, on Aaraonson’s blog.)
And, no, there won’t be “A Song I Like” section this time around. I still go jobless. Keep that in mind.
[Updated with a few cuts+some additions on July 12, 2012.]