Leaps in Science

Discussions on the philosophical foundations, assumptions, and implications of science, including the natural sciences.

Leaps in Science

Postby Marshall on October 5th, 2014, 2:26 pm 

The issue of "hypothesis formation" was raised by DLorde in the "Limits" thread. It's very interesting so I hope to start a new discussion specifically about that. Please contribute examples of conceptual leaps, if you can think of any you find especially intriguing or enlightening.

I want to stress that hypothesis formation is a kind of pattern perception which goes on all the time in human life not confined to scientific communities and sometimes can be mistaken. Perception of a pattern can be wrong or only approximately right. There may be a better pattern to explain the same behavior, to be found later. A pattern is not intrinsic to the data, seeing a pattern depends on cultural background and preparation, what the person BRINGS, their vocabulary of patterns (e.g. mathematics) that they are able to perceive in or apply to the data. And then there is the business of testing---which is also not confined to branches of science.

That said, I'm inviting us to consider scientific work as an EXAMPLE, of pattern perception, hypothesis formation. Where do the ideas come from? for the LEAPS that occasionally are made.

Re: Leaps in Science

Postby Marshall on October 5th, 2014, 2:54 pm 

It turns out that HEMIOLA in greek and SESQUIALTERA in Latin mean the same thing and both originally referred to PITCH ratios. the interval "fifth" refers to the DO-SOL interval because sol is the fifth (do re me fa sol). Etymologically both words mean "another half". You have one slice of pie and you add another half slice, so you have one-and-a-half slices, or 3/2 of a unit.

Sesquialtera was the term in medieval music for the 3/2 ratio. If one organ pipe is 3/2 the length of the other, the longer will play DO and the shorter will play SOL. If you touch the halfway point of a lute string (so it vibrates in halves) and then touch the 1/3 point (so it vibrates in thirds) you get the do-sol interval. Something singing sol is vibrating three times for every two times that something singing do is vibrating. It is vibrating 1.5 times faster. That is the sesquialtera.

Kepler discovered the Sesquialtera in the Solar System and this opened the door for newtonian dynamics.

The orbit period P and radius R of all the planets are related by P2 = R3

this pattern joins time and distance (the joining essential to dynamics)

Owleye had a post in the "Limits" thread about music opening people's minds.

In 1619, when Kepler stated what became known as his third law, he used a term from music to describe planetary motions.
The book, Harmonice Mundi, was published (amusingly) by the printer Johann Planck. It covers various topics, not just astronomy, in volume 5 he says (Wikipedia):
"Sed res est certissima exactissimaque quod proportio qua est inter binorum quorumcunque Planetarum tempora periodica, sit præcise sesquialtera proportionis mediarum distantiarum, … " (But it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialternate proportion [i.e., the ratio of 3:2] of their mean distances, … ")

Re: Leaps in Science

Postby Marshall on October 5th, 2014, 3:08 pm 

It could be we all know what I'm trying to say with this example: that something nontrivial can be going on when people see patterns in data and form hypotheses. and also that the mental process can turn out to have been logically in appropriate to the data, as in the case of Kepler, but no matter, he was prepared (by his culture and the mathematics of his time) to make the leap anyway.

The word hemiola comes from the Greek adjective ἡμιόλιος, hemiolios, meaning "containing one and a half," "half as much again," "in the ratio of one and a half to one (3:2), as in musical sounds."[1] The early Pythagorians, such as Hippasus and Philolaus, used this term in a music-theoretic context to mean a perfect fifth.[2]...

The words "hemiola" and "sesquialtera" both signify the ratio 3:2, and in music were first used to describe relations of pitch. Dividing the string of a monochord in this ratio produces the interval of a perfect fifth.

Hemiola is the ratio of the lengths of two strings, three-to-two (3:2), that together sound a perfect fifth.[16]

The justly tuned pitch ratio of a perfect fifth means that the upper note makes three vibrations in the same amount of time that the lower note makes two... The just perfect fifth can be heard when a violin is tuned: if adjacent strings are adjusted to the exact ratio of 3:2, the result is a smooth and consonant sound, and the violin sounds in tune.

From high to low, the violin strings go E, A, D, G.
D is a fifth above G, A is a fifth above A, and so on. Each step is a sesquialtera and that was a Medieval music term, a concept in Kepler's culture. Here is the kind of data he had in front of him to look at:

Code: Select all
Radii          Years
.72               .61
1.0               1.0
1.5               1.9
5.2              11.9
9.5              29.4

If you cube 9.5 and then take the square root, do you get 29.4? He recognized somehow that you do.
Let's paste 9.5^1.5 into google, and 5.2^1.5, and 1.5^1.5 ...pretty close.
.72^1.5 gives almost exactly .61
Some of the others are off by one in the tenths digit, like 29.3 instead of 29.4. Just roundoff.

Anyway take the number and multiply it by its own square root like asking for another HALF slice of pie. A sesqui altera. Half another, please.

What other example of "hypothesis formation" can you think of that you might like to share?

Re: Leaps in Science

Postby Faradave on October 5th, 2014, 6:56 pm 

If I take your meaning, I believe all the classic unifications would have to qualify as these leaps.

Newton's unification of celestial and terrestrial motion.
Maxwell's unification of electricity and magnetism.
Maxwell's & Boltzmann's unification of thermodynamics and mechanics in kinetic theory.
Einstein's unification of mechanics and electrodynamics in SR then GR.
Einstein's unification of space and time in spacetime.
Einstein's unification of inertial mass and gravitational mass in GR.
Dirac I believe gets most credit for electroweak unification.

There are others which should be added.
Certainly there are some separations to be considered as well. The separation of supernatural from natural for example, as a means of explaining observed phenomena.

Perhaps these are on too grand a scale?
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Re: Leaps in Science

Postby dandelion on October 7th, 2014, 10:52 am 

Patterns that might fit such patterns might include philosophical concepts based on multiplicity, or the breakthrough said to have occurred after Ouroboros day-dream?
(Edit: wiki links to Multiplicity (philosophy) and Kekule didn't work for me, and 'day-dream' is the wrong word, but not sure how to describe it.)
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Re: Leaps in Science

Postby Marshall on October 7th, 2014, 11:45 am 

A perfect example! the intuition of the molecular structure of BENZENE. The visual intuition breakthrough insight that it was a RING! That example hadn't occurred to me. Thanks.
Here is the ouroboros wiki-entry:

Here is the brief section they have with August Kekule's account of what occurred to him in 1872:
I'd like to quote it, since it is a vivid first-hand experience not everyone knows about:
I was sitting, writing at my text-book; but the work did not progress; my thoughts were elsewhere. I turned my chair to the fire and dozed. Again the atoms were gamboling before my eyes. This time the smaller groups kept modestly in the background. My mental eye, rendered more acute by the repeated visions of the kind, could now distinguish larger structures of manifold conformation: long rows, sometimes more closely fitted together; all twining and twisting in snake-like motion. But look! What was that? One of the snakes had seized hold of its own tail, and the form whirled mockingly before my eyes. As if by a flash of lightning I awoke; and this time also I spent the rest of the night in working out the consequences of the hypothesis.
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Re: Leaps in Science

Postby Marshall on October 7th, 2014, 12:41 pm 

Dandelion, your mentioning Multiplicity reminds me of Giordano Bruno who had the flash of insight that stars could have planets and there could be other inhabited solar systems. that really turned things upside down for people who had been thinking of our solar system as the central figure of the cosmic picture with the stars as trimmings on the fringe. an unfortunate thing happened to Bruno, but that is not the main issue here.

It also reminded me of the Leap of NON-CENTRALITY, which is in a way akin to multiplicity and that curiously goes back to a flash of insight in 250 BCE.

Aristarchus realized by a simple triangle visualization that the sun was much much farther than the moon.
He knew from looking at the curve of the earth's shadow on the moon that the earth and moon were about the same size (within a factor of 3 or so, same ball park)

And since the moon and sun LOOK the same size in the sky (same angular size), he reasoned that the sun was much larger than the moon (since it was so much farther away).

So he realized that the sun was MUCH LARGER THAN THE EARTH, and right there he grasped the NON-CENTRALITY of earth and conceived of the heliocentric model cosmos.

Later when Archimedes wrote a small booklet called "Sand-reckoner" for the entertainment and instruction of a friend's son he noted that "Aristarchus thought that the earth and planets orbit around the sun" or words to that effect. So he proposed that they calculate how many grains of sand it would take to fill us Aristarchus model of the cosmos (i.e. solar system). If I remember right that was how it went.

But the key LEAP was that TRIANGLE that Aristarchus saw around 250 BCE on an afternoon when there was a precise half moon. That was what made him realize that the sun was so much farther than the moon. I better get a cup of coffee and explain in another post. You may be familiar with this and it may have been discussed here at SCF earlier, I don't remember, but it's quick, so no harm repeating.

Re: Leaps in Science

Postby owleye on October 7th, 2014, 1:08 pm 

Taking a more prosaic orientation to this question, it might be useful to consider how a criminal detective comes to conclusions from her investigations. The reasoning process comes up with plausible suspects which might be culled through further research. However, though the approach is cumulative, and presumably gets us closer to "whodunit", the truth of the matter is problematic if it concludes that it was A, but later it turns out that it was B. (Note my wife and I just saw the movie "Gone Girl", which emphasizes some of the difficulties of determining truth from evidence.)

Re: Leaps in Science

Postby Marshall on October 7th, 2014, 1:32 pm 

Hi Owl! Many people know the Aristrarchus half moon story, I suspect, but it's worth repeating.

About a week after the new moon there is the "first quarter" or half moon and you see it high in the sky late in the afternoon, going on towards sunset. I like to walk up on an openspace hill near sunset and I see the moon then, an exact half, at the high point of its arc with its curve to the west (to my right) where the sun is beginning to set, and its flat edge on my left, to the east.

Then I know that earth, moon, and sun form a right triangle EMS, with the 90 degree angle at M.

So I estimate the angle where I stand, at E, and I judge it is not quite 90 degrees, something like 89 degrees.

I have a walking stick to help me climb the dirt trail up the hill and with my stick in my right hand I point to the setting siun while with my left arm I point at the precise half moon which is near the centerline of the sky. The angle between those two lines ES and EM is just a bit less than 90. Something like 89.

At this juncture, Aristarchus moved his viewpoint to the sun! (This is the cool thing, he moved his perspective.)

He realized that if he was at S and looked back at E and M the angle between them would have to be very small, something like ONE DEGREE. He realized that the triangle was very SKINNY.

The angle at the sun, between the earth and moon, had to be the amount by which the angle I measured by pointing at the two things was less than 90 degrees. If it was 89 then the angle at S had to be 1.

So he figured out, by simple trig, that the sun was a lot farther away than the moon. ES was a lot longer than EM in that triangle. He estimated the ratio as at least 10, or 12, I forget exactly.

So he realized the non-centrality and got a glimpse of the heliocentric picture.

Aristarchus started people measuring ratios of sky distances, and eventually Kepler connected the distances to the TIMES that orbits take. I hope there's a party for Johannes Kepler in 2019.
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Re: Leaps in Science

Postby owleye on October 8th, 2014, 1:58 pm 


Though not directly responding to my post, I found it very enlightening. The basic element is that science is capable of coming up with true statements about the world. In this case, that the sun is much farther away from us (the earth) than is the moon. And even though this hadn't been common knowledge at the time, and apparently didn't become common knowledge for some time, it now is common knowledge. Indeed, it may be one of the key separators that mark the distinction between the east and the west (or at least the cultures in an around what was called "the near east").

Overnight I've given some thought to this train of thought, particularly because there are some issues involved in what science achieves. Instrumentalists tend to think that only a pragmatic truth is available to it (i.e., whatever works) or that truth itself is not in the cards. I can imagine theorists see it a bit differently though.

In any case it is worth while I think to take a look at some of the assumptions (probably) made by Aristarchus.

1. The phases of the moon are produced by the light from the sun. (Probably common knowledge as I suspect it was known by almost every culture that gazed upward.)

2. That Euclid's geometry (not precisely what we mean by Euclidean geometry, but close enough) was applicable and allowed the portrayal of distances in accord with, for example, the Pythagorean theorem. (This has been speculated to be the result of the geography of Greece, respecting navigational principles involved in island mapping or other charting needs. Possible, I suppose.)

3. The light that is reflected off the moon and into our eyes and is subsequently perceived by us takes place as if it occurred all at the same time.

4. Appearances (what we perceive) are steadfastly indicative of what's true (in defiance of Plato' shadowy world, I suppose).

5.. Most importantly, that empirical means can be used to obtain true statements, one that implies the concept of measurement (as in geometry), one that can be applied by anyone to anything -- thus has an objective aspect to it.

There may be and probably are more assumptions, but given these five, we might today conclude that western science benefitted from having them as a basis for their success. The truth being claimed might not seem like a huge one because it's not saying that Euclid's geometry is in fact the true geometry, or that light must travel at an infinite speed, but that there is a way of thinking about space that allows one to draw these kinds of truths. There is a sense in which we can arrive at true statements by declaring them in a certain way, and from which we are able to make progress by examining the assumptions.

Re: Leaps in Science

Postby Marshall on October 8th, 2014, 5:28 pm 

Hi Owleye, thank for the careful thinking-through. I'm glad you find Aristarchus story interesting. As I recall Anaximander somehow got the notion that the sun was not quite twice as far. Something like in ratio 11 to 7 or in ratio 7 to 4.

He was back around 550 BCE (born around 600 BCE) and Aristarchus came 300 years later, around 250BC.

they were both Ionian greeks living in what is now turkey, coastal cities and nearby islands. A lot of islands are just offshore from Aegean coast Turkey.

I sometimes wonder if it wasn't the islands that made those Ionians wonder "how far" so much.

An island is something you can see, that you wonder how far about. It's always on your mind because you see it every day. Pythagoras grew up on Samos island which is just off shore from Miletus, where Thales and Anaximander lived.

It would be easy for people living in an archipelago with so many islands within sight of each other to get interested both in navigation and in geometry.

Geometry is so important! Your post is partly about that.

The great conceptual leap that Anaximander made, sometime around 550BCE, was that DOWN IS RELATIVE.

So the earth does not need supports to hold it up! there is no absolute up, that turtles or giants need to hold the earth. Down is towards the earth center so it depends where you are.

So since the earth doesn't need supports holding it up THE SUN MOON AND STARS CAN GO UNDER THE EARTH WITHOUT BUMPING ANYTHING.

This is so beautiful. It is truly a great leap of insight. So then the idea of a sphere of stars rotating could get started in people's minds. The sun could ride on a wheel around the earth because there was nothing to get in the way of the wheel. And Anaximander even began this mechanistic thought process and estimated the radii of the sun wheel and the moon wheel. The sun's wheel was bigger but not by a whole lot.

BTW there is a distinction which you are probing which is AFAIK not resolved in a hard and fast way. More resolved by people's USAGE or community consensus i.e. common sense (or sense of words we as a community have in common). This is the distinction between facts and theories.

Facts are accepted as true. Theories are based on facts and offer to EXPLAIN facts, and theories are treated as provisional and subject to improvement. A theory is something with a "domain of applicability" outside of which you are not advised to trust it. Like GR is not applicable in situations with extreme energy density and extreme curvature, because it blows up.

Today we accept as FACT that the sun is much farther than the moon. But maybe for Aristarchus in 250 BCE it was part of his theory to explain the angle in the sky between the half-moon and the sun. And subject to revision. Indeed people attacked and rejected his heliocentric idea. it was a theory. They weren't always consistent.

So there seems to be no automatic sharp distinction between theory (provisional explanation) and fact (accepted data which theory should explain). The same statement can be either one depending on the historical circumstances.
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Re: Leaps in Science

Postby owleye on October 9th, 2014, 7:54 am 


I think it's reasonable to say that common knowledge is factual -- fact-based, even if at some point in its history it wasn't known to be. And one would consider it a fact if it were true. If this were all there was to it, though, we would have to explain why what was believed before it was accepted as true wasn't true. To tackle this, I think, one needs to make a case first that prior beliefs were not based on a sufficiently solid methodological foundation (which emphasized measurement) while the newer is so based and second that the truth of the new version isn't overstated to refer to every facet of how its theoretical structure came to be accepted. Theory and fact are related in such a way that the one is supposed to explain the other. The question about truth, though, isn't so much that it is a true explanation, but that it is an adequate explanation. Darwin's theory of the evolution of life explains the facts of life. And it allows us to draw true inferences -- e.g., that offspring of certain life forms to include humans are formed by sexual reproduction involving the union of (potentially corrupted) parental gametes of the opposite sex (subject to viability constraints associated with pairing up) and in total represents a randomized combination of traits of each of the parents. This is off the top of my head and probably inadequate, but I wished to express something that is common knowledge and true, and as well based on solid science yet might have been believed otherwise prior to that science. There are I would think thousands, if not millions of true statements of this sort, and many ready at hand in the face of objectors to Darwin's theory. Perhaps, though, I'm mistaken and it's not so easy to give these kind of true statements about life, so what we do is declare Darwin's theory a fact. It would be my hope that this isn't necessary, but given that this is science's general practice, and biologists are in a better position than I am, perhaps it's too much to ask to find true statements that are common knowledge and were arrived at through a solid evidentiary based empirical science that heretofore wasn't common knowledge to thwart naysayers who still think of it in the old way. Perhaps we need another thousand years or more to get to that point.

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