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Friday, January 22, 2010

Does Math Lead to Understanding?

In "SolderSmoke -- The Book" I describe the quest for deep understanding of the circuits that we build and use. There is some discussion in the book of the role of mathematics in this quest. A while back a reader e-mailed me on this subject. In the hope of stimulating a discussion, I'll present the key paragraph from that e-mail here (the author will, for now, remain anonymous):

I appreciate your quotes from Feynman, Asimov, etc. about not
really being able
to fully understand everything. As a math teacher
I can say that one of the
biggest misunderstandings about math
is that it "explains" the phenomena of
physics and engineering.
(Science and
math teachers are notorious for saying to a student
who has just asked a "why" question things like, "well the
math is
a little bit more complicated than what you can handle right now.
Wait until
you have had a year or so of calculus.") In reality it's
the exact opposite!
The math equations actually hide the answers.
They are very good at accurately
describing phenomena, or at
predicting what will happen next, but they can never
answer the
question of why one equation works and another does not. We
get very
comfortable with allowing the familiar math equations
to hide our inability to
really answer the "whys."

This really resonated with me. In my effort to get a better grasp of mixer theoy a lot of people seemed to be simply pointing me to the trig equations, and equating a knowledge of those equations with an understanding of how the mixer circuits really work.

Of course, I don't mean to be anti-math here, but I thought the e-mail on the limits of mathematics was very interesting.
In "Empire of the Air" Tom Lewis wrote, "At Columbia, Edwin Howard Armstrong developed another trait that displeased some of the staff and would annoy others later in life: his distrust of mathematical explanations for phenomena of the physical world. All too often he found his professors taking refuge in such abstractions when faced with a difficult and seemingly intractable conundrum... Time and again as an undergraduate at Columbia, Armstrong had refused to seek in mathematics a refuge from physical realities."

9 comments:

  1. By looking at the “softer sciences” rather then the “harder” sciences it is easier to under the role of mathematics in explaining natural phenomena. In my graduate studies of ecology and cultural anthropology I have learned mathematical models that are used to represent human or animal foraging behavior, predator prey dynamics, and other examples of human and natural phenomena. When applied to naturally occurring systems, these models have limited predictability, and accuracy. Because of this trend many say that the models are really poor. I feel that the models are not wrong. When a model is constructed assumptions about reality are made. Cultures and ecosystems contain factors that are stochastic and/or non random that violates the model’s assumptions. So, I think that mathematics helps scientist in understanding simple patterns seen in nature, but for us to understand the more complex patterns of reality we need to understand the limitations of math, so that it does not become the crutch that it is in physics, and some other fields.

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  2. I agree. Despite being an (amateur) programmer I have completely failed to understand how SDR or DSP work at all. All the explanatory articles are riddled with incomprehensible equations, which may explain what happens to a scientist or a mathematician but not to a layman.

    Perhaps due to some deficiency of my brain, I have never been able to grasp abstract concepts. All of my understanding of anything is based on my having a mental picture of how it works. Mathematical formulae mean nothing to me at all, and if something can't be explained without them I will never understand it.

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  3. Armstrong wrote an article in 1948-50 addressing this very issue. He strongly suggested that while mathematics has its place experiment and observation were MORE important. Perhaps a one of the Solder Smoke fans has that article and could make it available??
    72/73
    WR9H
    Herb

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  4. I believe that mathematics is important in physics not because it "explains" things, but because it is the method by which we create models of the universe about us, and test these methods against reality. When we use (say) the small signal approximations for figuring out the behavior of transistor circuits, we aren't really "explaining" anything about transistors. We have a model which has proven useful that allows us to accurately determine the behavior (within the limits of the model) of a system that we couldn't really understand without using the model.

    Lots of people don't seem to understand math very well. It's hard to know whether that's due to a lack of basic aptitude or a lack of adequate instruction. I tend to think that it is more often the latter. Concepts of use to the radio amateur, such as certain trig identities and the FFT are often presented in the most obtuse manner imaginable, and lend intuition often only in retrospect.

    On the other hand, what is the alternative? Bill has podcasted about his attempt to homebrew amplifiers before. They worked, so it might be reasonably claimed that he "understood" amplifiers. But working through the math in EMRFD using LTSpice, (along with practical experimentation) he was able to actually design an amplifier that met some basic performance requirements. He could take a simulated circuit and build it with some confidence that it would work.

    I think that math is essential to our understanding of physics. Galileo's experiments with gravity were a revolution in science, precisely because he was the first to develop a mathematical model for gravitation. It wasn't just qualitative: it was quantitative and predictive. Without it, we are fumbling in the dark, or at least craftsmen rather than engineers.

    Thanks for the food for thought, Bill.

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  5. I don't think Math directly leads to understanding, but without it I know all we have is superstition and chaos. Engineering and Science requires a strict system like mathematics to describe models of reality.

    Maths can be used to describe and model the observed behaviour of this universe unreasonably well. Why I don't know. I don't think anyone really knows why Math works, but unfortunately it is the only tool we have of such power.

    Maths is however a rather blunt instrument. Equations might describe a modelled system, but true wisdom is difficult to extract from a terse mathematical statement. The mathematician spends most of their life developing (or the truly gifted are born with) the ability to intuitively know the implications of mathematical constructs. Sadly I don't have much of that ability, few do, but like most I can use Maths in a basic fashion to achieve real goals.

    A classic example is electromagnetic radiation. Maxwell's Equations at first look are rather incomprehensible. Each was formulated by different researchers, working on problems that seemed to have no direct connection, but buried in them are the implications that they can be solved for transverse propagating modes we now know describes light, radio, x-rays, etc. They also imply some things about charge conservation, the non-existence of magnetic charge (monopoles), etc. To see those implications from the equations takes a natural talent or years of learning and experience.

    In your mixer example, the trig identity that implies sum and difference product formation is rather unlikely to be discovered by casual inspection.

    Everyone has different levels of mathematical expertise and intuition. I suspect a lot of the common belief that "Math is hard" comes from how it is taught. Past a certain point the uninitiated would be utterly lost when faced with some deep mathematical construct. With suitable introduction most things are actually fairly straight forward and perfectly logical. Mathematics is a vast discipline, with nomenclature that has grown from the input of thousands of years of work from countless different people with wildly different cultural backgrounds and natural languages. It is no wonder that many parts are not very welcoming to the newcomer. Most professional mathematicians are specialists, and outside their own areas of interest they too are rather lost without first reading the basics in a new area.

    One of my Maths lectures once told me "Mathematics is not a spectator sport". This is very true in my experience. The only way to develop mathematical intuition is to play the game, use maths as much as possible. The more you use it, the easier it gets and the more "ah ha" moments you'll have, when it clicks and you see why something was structured the way it is, and what deep connections it has to other concepts.

    Regards,
    Alan

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  6. I'm with Alan -- maths is definitely not a spectator sport.

    The use of maths in electronics has, in my opinion, two goals. The first is to allow you to design circuits that behave in the way you want by modeling physical phenomena, such as transistor currents etc.

    The second goal is to provide a theoretical framework that allows you to find the ideal circuit behavior you require in order to perform a specific task.

    An example of the latter is the trigonometric mixer formula that Bill refers to. The most important aspect of the formula is not the use of sin() or cos() but the multiplication.

    The formula shows that, if we want to move a single tone to another frequency, we can do so by multiplying the tone by a second tone.

    Therefore, any circuit that can perform a multiplication of two signals, no matter how crude, will exhibit a mixing function to some extent.

    Furthermore, the best mixer circuit is the circuit that performs a multiplication of two signals most accurately.

    Is it really possible to understand this? I'm not sure. Perhaps understanding involves convincing one's self that what the mathematics show, is actually "true", by experiment or other methods.

    There are certainly many levels of understanding. I believe that the more ways you have of looking at a circuit, mathematical or otherwise, the better your understanding is and the more "ah ha" moments you will have.

    I used to dislike maths but now I can't go without it. If you can keep a secret: maths are actually fun -- but don't tell anyone I said that!

    73,
    Niels PA1DSP.

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  7. Alan Yates wrote: "Maths can be used to describe and model the observed behaviour of this universe unreasonably well. Why I don't know."

    Math works well in describing our observable universe, because we have created a mathematical system that corresponds to our observations. There is nothing to prevent us from creating a mathematical system where 1+1=3, but that doesn't correspond to our observable reality, so we didn't built a mathematical system that permits 1+1=3.

    A good book to understand mathematical systems is Douglas Hofstadter's "Godel,Escher, Bach". This is a book, I must admit, that I have dived into numerous times over the years, but have never made it to the end. But it is a very inspiring text on math, art and music, and how they are interconnected.

    But back to Bill's original frustrations.... Math is poorly taught, probably because it is taught by mathematicians, for whom the math is an end onto itself. The rest just don't really get it either. It used to be that physicists and engineers relied on math to model observed physical phenomenon. It gave them a language to describe what they measured and observed, the advantage being that once you got the 'language' correct, you could extend the 'language' to describe similar phenomenon which you had not directly observed or measured.

    Most of us prefer a mechanical model of the world, as we can visualize it in terms of things we can physically experience. In electronics, this can be difficult. Not impossible, but very difficult. Math is sometimes a shortcut to creating a very complex mechanical allegory.

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  8. Mathematics is a language that can be used to describe things. It can describe the way things are, how they work, and (given some assumptions) how they probably will work in different situations. Like any language, it doesn't magically explain "why" something works - that is up to the person using the language to do that!

    I agree with several posters that Math is poorly taught. As an Applied Mathematics Major, I spent many years tutoring Math, and have universally found that even the students having the most problems were able to understand and apply the math (i.e. get A's on tests) when it was explained to them in a manner that made sense to *them*. The key was usually showing them how the Math related to real-world things that they were already familiar with.

    Just my 2 cents.

    Greg
    KJ4IIN

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  9. Mathematics is the "language" of science and engineering but unfortunately the naive are led to believe that the math drives the engineering and it's actually the other way around.

    This has had DISASTROUS effect on our undergraduate engineering programs because there is much less emphasis on DOING the engineering (building things that have some use and worth) and more emphasis is placed on "analysis" which is French for TALKING about doing something.

    The end results are all around you-from cars with faulty brake systems to shuttles that blow up. We've so convinced ourselves that because the "math says it works" (or the computer program for that matter), that we've lost the creative and intuitive side. This means we build things that over-heat, build items that can stand up to normal stresses such as accidental dropping to the floor, or are so complicated that users can't figure them out.

    One of the greatest electrical engineers EVER was a guy by the name of Oliver Heaviside. Heaviside had developed an algebraic means of solving differential equations through a means of going from the time domain to the frequency domain and changing the problem from that of complex exponentials to that of algebra.

    Mathematicians during his time HATED the idea as they had no tool to prove that it worked. Today, the "Heaviside Transform" (called the "Laplace Transform" in our politically correct books) is widely used in all forms of engineering problems.

    When faced with criticisms by the mathematicians, Heaviside made the following comment "I don't need to understand my digestive system to enjoy a good meal".

    I think that sort of comeback is terrific. It comes from a guy who was a TINKERER 1st and a purist 2nd.

    Even a fellow like Feynmann (who was a good math guy in addition to being a physicist) looked for ways to simplify his ideas. You'll find that the TRULY gifted and bright people of the world can make the complicated look simple. It's only when we get a hyper-educated specialist involved that we make things "complicated" because that allows for "experts".

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