@gregeganSF In a certain way the chart is curious. Because you can view sets as a structure. There is base collection (classes) something more (the belonging relation) axioms.
It is not very different from groups: there is a base collection (sets) something more (an operation) and axioms.
When Maxwell realized in 1862 that light consists of waves in the electromagnetic field, why didn't anyone try to use electricity to make such waves right away? Why did Hertz succeed only 24 years later?
According to 𝘛𝘩𝘦 𝘔𝘢𝘹𝘸𝘦𝘭𝘭𝘪𝘢𝘯𝘴:
"Since he regarded the production of light as an essentially molecular and mechanical process, prior, in a sense, to electromagnetic laws, Maxwell could elaborate an electromagnetic account of the propagation of light without ever supposing that ether waves were produced purely electromagnetically."
In 1879, a physicist named Lodge realized that in theory one could make "electromagnetic light". But he didn't think of creating waves of lower frequency:
"Send through the helix an intermittent current (best alternately reversed) but the alternations must be very rapid, several billion per sec."
He mentioned this idea to Fitzgerald, who believed he could prove it was impossible. Unfortunately Fitzgerald managed to convince Lodge. But later he realized his mistake:
"It was FitzGerald himself who found the flaws in his "proofs." He then proceeded to put the subject on a sound theoretical basis, so that by 1883 he understood quite clearly how electromagnetic waves could be produced and what their characteristics would be. But the waves remained inaccessible; FitzGerald, along with everyone else, was stymied by the lack of any way to detect them."
In 1883, Fitzgerald gave a talk called "On a Method of Producing Electromagnetic Disturbances of Comparatively Short Wavelengths". But he couldn't figure out how to 𝘥𝘦𝘵𝘦𝘤𝘵 these waves. Hertz figured that out in 1886.
The "experience" of Herz was a mere chance observation. Here is a schema (1:45) What he observed was that every time there was a spark in the emitter, there was also a spark in the receptor. But he understood that it validated the transmission of electromagnetic fields and then Maxwell's ideas.
(You can activate the subtitling and the translation in the video but it is not always correct.)
@johncarlosbaez@BashStKid "Not seeing any way to build an apparatus to experimentally test this, Hertz thought it was too difficult, and worked on electromagnetic induction instead."
He'd given up. But it explains why he knew Maxwell's theories so intimately.
@johncarlosbaez@BashStKid "he was experimenting with a pair of Riess spirals when he noticed that discharging a Leyden jar into one of these coils produced a spark in the other coil. With an idea on how to build an apparatus"
It looks like the "chance observation" I described. Except that it doesn't use the equipment in the diagram.
@johncarlosbaez@BashStKid "The antenna was excited by pulses of high voltage of about 30 kilovolts applied between the two sides from a Ruhmkorff coil. He received the waves with a resonant single-loop antenna with a micrometer spark gap between the ends."
This time it corresponds to the diagram but it's an experiment set up after the involuntary observation described in the message above.
@johncarlosbaez@BashStKid "Hertz did produce an analysis of Maxwell's equations during his time at Kiel, showing they did have more validity than the then prevalent "action at a distance" theories."
It was what I thought tonight: was it a "action at a distance" or a "transport"?
Fun article by John Psmith featuring some ferociously competitive mathematicians and physicists. A quote:
.....
In the 1696 edition of Acta Eruditorum, Johann Bernoulli threw down the gauntlet:
"I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.
Given two points A and B in a vertical plane,
what is the curve traced out by a point acted on only by gravity,
which starts at A and reaches B in the shortest time."
This became known as the brachistochrone problem, and it occupied the best minds of Europe for, well, for less time than Johann Bernoulli hoped. The legend goes that he issued that pompous challenge I quoted above, and shortly afterward discovered that his own solution to the problem was incorrect. Worse, in short order he received five copies of the actually correct solution to the problem, supposedly all on the same day. The responses came from Newton, Leibniz, l’Hôpital, Tschirnhaus, and worst of all, his own brother Jakob Bernoulli, who had upstaged him yet again.
(1/2) (The fun part about Newton comes in part 2.)
@BartoszMilewski Same problem with classical logic. They give real-life examples. Any critical mind is quick to take exception. Especially young people.