Poincaré's Take on Theories

I've always been interested in the ideas of mathematician and philosopher Henri Poincaré. While I have not read any of his philosophical work, his ideas as summarized by others have always resonated with me. Here's an excerpt from the Internet Encyclopedia of Philosophy on Poincaré's thoughts concerning the relationship between observations and theories (italics denote my emphasis).

According to Poincaré, although scientific theories originate from experience, they are neither verifiable nor falsifiable by means of the experience alone. For example, look at the problem of finding a mathematical law that describes a given series of observations. In this case, representative points are plotted in a graph, and then a simple curve is interpolated. The curve chosen will depend both on the experience which determines the representative points and on the desired smoothness of the curve even though the smoother the curve the more that some points will miss the curve. Therefore, the interpolated curve — and thus the tentative law — is not a direct generalization of the experience, for it ‘corrects’ the experience. The discrepancy between observed and calculated values is thus not regarded as a falsification of the law, but as a correction that the law imposes on our observations. In this sense, there is always a necessary difference between facts and theories, and therefore a scientific theory is not directly falsifiable by the experience.

Poincaré did not likely consider systematic errors in the reason for why observations did not match a theory, so it is interesting that a theory that does not match the experimental data precisely is not necessarily wrong.

How does the uncertainty principle tie into this argument?

Nuclear power-yes or no?

There's a good article from Damian Carrington's Environment Blog in The Guardian today that you can access here: http://www.guardian.co.uk/environment/damian-carrington-blog/2011/apr/21/chernobyl-nuclear-power-fukushima. The article addresses the timely question of whether society should invest in building new nuclear reactors. Rather than argue for a specific viewpoint, Carrington instead offers five important questions that, when considered personally, assist in determining one's own stance towards expanding mankind's arsenal of nuclear power plants.

I must add that I do appreciate this type of blog post. It seems to me that readers may be more focused on the post's contents when they are actively engaged in the thought process as opposed to interpreting the author's opinions. Should I practice this type of blogging? :)

Amplitude and phase: what are they really?

For sinusoidal signals, the idea of amplitude and phase are pretty clear. The amplitude is one half of the peak to valley distance of the signal and phase coincides with the position of the signal at a point in time or space. By signal, I mean any physical quantity that is undergoing simple harmonic motion, e.g. a mass on a spring. In optics, the amplitude and phase of plane waves are well-defined as well: the amplitude is related to the irradiance and the phase typically refers to the position of the wavefronts with respect to some reference.

What's often taken for granted by new physics students (and even experienced ones) is that plane waves and sinusoidal signals are not real. Any real signal must be of finite duration; a sinusoid extends to infinity both forwards and backwards in time. As a result, the amplitude and phase of real signals become somewhat ambiguous.

The two concepts are retained in practice, however, because they simplify the interpretation of many physical situations, such as the effects of a filter on a signal or the output of an optical interferometer or two-slit experiment. A real signal s(t) has an analytic representation z(t) that consists of s(t) plus the signal's Hilbert transform times the imaginary number, j (I was trained as an engineer, but consider it the same as i used by physicists and mathematicians). Two things should be noted here: 1) the analytic signal is unique, and 2) the analytic signal is complex; it has a real and imaginary part. Like all complex numbers, it can also be represented by a magnitude and phase angle. The magnitude is often denoted the amplitude and the phase angle is thought of as the phase of the signal.

Unfortunately, the amplitude and phase of the analytic signal can not always be directly correlated to physically meaningful quantities. For frequency modulated signals, the time-varying amplitude and phase are independent only so long as their individual spectra are non-overlapping. Furthermore, even if the time-varying amplitude and phase are spectrally separate, it is very counter-intuitive (at least to me) to be able to represent one function of time (the real signal) with two functions of time. This last point I believe is related to the Kramers-Kronig relationship between the real and imaginary parts of a signal. So, while a time-varying amplitude and phase can represent a real modulated signal and simplify interpretation, the only thing about the signal with any physical significance is what can be measured, such as current or voltage.

Most of this post comes from thoughts I've had recently concerning how casually we in optical sciences use the terms amplitude and phase. Once an optical signal becomes incoherent, either in space or time, its amplitude and phase are said to vary randomly. This statement only really makes since once one considers that amplitude and phase are purely theoretical constructs for describing real data and that one must not place too much physical significance in their meaning.

I also learned a lot and reproduced some information from this very good article by Boashash in the Proceedings of the IEEE: http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=135376&tag=1.

More on mental calculations

Just a quick post today. I found this article in Nature Neuroscience from Wikipedia's page on mental calculation.  The abstract follows:

Calculating prodigies are individuals who are exceptional at quickly and accurately solving complex mental calculations. With positron emission tomography (PET), we investigated the neural bases of the cognitive abilities of an expert calculator and a group of non-experts, contrasting complex mental calculation to memory retrieval of arithmetic facts. We demonstrated that calculation expertise was not due to increased activity of processes that exist in non-experts; rather, the expert and the non-experts used different brain areas for calculation. We found that the expert could switch between short-term effort-requiring storage strategies and highly efficient episodic memory encoding and retrieval, a process that was sustained by right prefrontal and medial temporal areas.

It seems that individuals who are naturally gifted at mental calculation possess a sort of encoding mechanism for rapid recall of previously acquired mental facts.

Fortunately, I believe our brains are not hard-wired but able to adapt to imposed stresses and routines so that even non-prodigies can become proficient at performing tasks such as these.

http://en.wikipedia.org/wiki/Mental_calculation
http://www.nature.com/neuro/journal/v4/n1/full/nn0101_103.html

Science and cooking online

Those who know me are aware that I am a big science buff. Those who know me even better know that I also am an avid cook and baker.

Harvard gave a lecture series this past fall on science and cooking and have graciously posted the lectures online. I recommend taking a look here.

Topics include thermodynamics, colloids, and sous vide cooking.

Mentat training

For Christmas I received a "Lightning Calculation" calendar. It's a wall calendar that explains techniques for performing mental calculations, such as finding what day of the week a date fell on and performing fast multiplications. In addition, there are historical bits on famous individuals who have been able to perform incredibly complicated mental calculations. I've spent at least half an hour in my office every morning reading through it and trying the practice problems (of which there are hundreds, if not over a thousand).

Something I've learned is the anchor method for multiplication of two, two-digit numbers. It may seem complicated at first, but it's actually incredibly powerful. First consider the product to be found as a product of two sums, each one "anchored" to a nearby round number. This product can be written as

(a + c)(a + d) = a^2 + ac + ad + cd
(a + c)(a + d) = a(a + c + d) + cd

where a is the anchor. From the right-hand-side of the last line above, the anchor is multiplied by itself with the sum or difference terms c and d. Then the correction term cd is added or subtracted, depending on whether the signs of c and d were the same or not, respectively.

For example, consider the product 17 * 18. This can be written as (20 - 3) * (20 - 2) = 20(20 - 3 - 2) + (2 * 3) = 306. The algorithm is easy to carry out because the first product is 20 * 15 = 300. Cool stuff.

The information for the calendar can be found here: http://myreckonings.com/wordpress/2010/11/22/a-2011-%E2%80%9Clightning-calculation%E2%80%9D-calendar/. It's very nicely done and I highly recommend it for all those mentats in training =)

More absolutes

Yesterday I wrote on the significance of relative and absolute measurements, concluding that relative measurements were in some sense less arbitrary than absolute ones because they do not depend on the definition of a physical constant. I claimed that this dependence is the practical problem with absolute measurements.

Further reflection has led me to believe that this is not the practical problem that has presented itself many times over during the course of my studies. Rather, the practical problem is that an absolute measurement is incredibly sensitive to the manner in which it is performed. Many parameters that are found in physical theories simply do not take account of the limitations in a measurement, such as integration times and nonlinearities in measuring devices. It is extremely difficult to extract a number from a measurement performed under realistic constraints.

Fortunately, any condition that ruins the agreement between a parameter obtained from a measurement and one obtained from calculation will be present amongst many different measurements. The effects of these conditions will essentially "divide out" under comparison, leaving only the signature of any independent variables that were changed between the measurements. Therein lies the strength of conclusions drawn from relative measurements.