Heat vs. Motion

by Eliezer Yudkowsky5 min read1st Apr 200870 comments

34

PhysicsMap and Territory
Personal Blog

After yesterday's post, it occurred to me that there's a much simpler example of reductionism jumping a gap of apparent-difference-in-kind: the reduction of heat to motion.

Today, the equivalence of heat and motion may seem too obvious in hindsighteveryone says that "heat is motion", therefore, it can't be a "weird" belief.

But there was a time when the kinetic theory of heat was a highly controversial scientific hypothesis, contrasting to belief in a caloric fluid that flowed from hot objects to cold objects.  Still earlier, the main theory of heat was "Phlogiston!"

Suppose you'd separately studied kinetic theory and caloric theory.  You now know something about kinetics: collisions, elastic rebounds, momentum, kinetic energy, gravity, inertia, free trajectories.  Separately, you know something about heat:  Temperatures, pressures, combustion, heat flows, engines, melting, vaporization.

Not only is this state of knowledge a plausible one, it is the state of knowledge possessed by e.g. Sadi Carnot, who, working strictly from within the caloric theory of heat, developed the principle of the Carnot cycle—a heat engine of maximum efficiency, whose existence implies the second law of thermodynamics.  This in 1824, when kinetics was a highly developed science.

Suppose, like Carnot, you know a great deal about kinetics, and a great deal about heat, as separate entities.  Separate entities of knowledge, that is: your brain has separate filing baskets for beliefs about kinetics and beliefs about heat.  But from the inside, this state of knowledge feels like living in a world of moving things and hot things, a world where motion and heat are independent properties of matter.

Now a Physicist From The Future comes along and tells you:  "Where there is heat, there is motion, and vice versa.  That's why, for example, rubbing things together makes them hotter."

There are (at least) two possible interpretations you could attach to this statement, "Where there is heat, there is motion, and vice versa."

First, you could suppose that heat and motion exist separately—that the caloric theory is correct—but that among our universe's physical laws is a "bridging law" which states that, where objects are moving quickly, caloric will come into existence.  And conversely, another bridging law says that caloric can exert pressure on things and make them move, which is why a hotter gas exerts more pressure on its enclosure (thus a steam engine can use steam to drive a piston).

Second, you could suppose that heat and motion are, in some as-yet-mysterious sense, the same thing.

"Nonsense," says Thinker 1, "the words 'heat' and 'motion' have two different meanings; that is why we have two different words.  We know how to determine when we will call an observed phenomenon 'heat'—heat can melt things, or make them burst into flame.  We know how to determine when we will say that an object is 'moving quickly'—it changes position; and when it crashes, it may deform, or shatter.  Heat is concerned with change of substance; motion, with change of position and shape.  To say that these two words have the same meaning is simply to confuse yourself."

"Impossible," says Thinker 2.  "It may be that, in our world, heat and motion are associated by bridging laws, so that it is a law of physics that motion creates caloric, and vice versa.  But I can easily imagine a world where rubbing things together does not make them hotter, and gases don't exert more pressure at higher temperatures.  Since there are possible worlds where heat and motion are not associated, they must be different properties—this is true a priori."

Thinker 1 is confusing the quotation and the referent.  2 + 2 = 4, but "2 + 2" ≠ "4".  The string "2 + 2" contains 5 characters (including whitespace) and the string "4" contains only 1 character.  If you type the two strings into a Python interpreter, they yield the same output,—> 4.  So you can't conclude, from looking at the strings "2 + 2" and "4", that just because the strings are different, they must have different "meanings" relative to the Python Interpreter.

The words "heat" and "kinetic energy" can be said to "refer to" the same thing, even before we know how heat reduces to motion, in the sense that we don't know yet what the reference is, but the references are in fact the same.  You might imagine an Idealized Omniscient Science Interpreter that would give the same output when we typed in "heat" and "kinetic energy" on the command line.

I talk about the Science Interpreter to emphasize that, to dereference the pointer, you've got to step outside cognition.  The end result of the dereference is something out there in reality, not in anyone's mind.  So you can say "real referent" or "actual referent", but you can't evaluate the words locally, from the inside of your own head.  You can't reason using the actual heat-referent—if you thought using real heat, thinking "1 million Kelvin" would vaporize your brain.  But, by forming a belief about your belief about heat, you can talk about your belief about heat, and say things like "It's possible that my belief about heat doesn't much resemble real heat."  You can't actually perform that comparison right there in your own mind, but you can talk about it.

Hence you can say, "My beliefs about heat and motion are not the same beliefs, but it's possible that actual heat and actual motion are the same thing."  It's just like being able to acknowledge that "the morning star" and "the evening star" might be the same planet, while also understanding that you can't determine this just by examining your beliefs—you've got to haul out the telescope.

Thinker 2's mistake follows similarly.  A physicist told him, "Where there is heat, there is motion" and P2 mistook this for a statement of physical law:  The presence of caloric causes the existence of motion.  What the physicist really means is more akin to an inferential rule:  Where you are told there is "heat", deduce the presence of "motion".

From this basic projection of a multilevel model into a multilevel reality follows another, distinct error: the conflation of conceptual possibility with logical possibility.  To Sadi Carnot, it is conceivable that there could be another world where heat and motion are not associated.  To Richard Feynman, armed with specific knowledge of how to derive equations about heat from equations about motion, this idea is not only inconceivable, but so wildly inconsistent as to make one's head explode. 

I should note, in fairness to philosophers, that there are philosophers who have said these things.  For example, Hilary Putnam, writing on the "Twin Earth" thought experiment:

Once we have discovered that water (in the actual world) is H20, nothing counts as a possible world in which water isn't H20.  In particular, if a "logically possible" statement is one that holds in some "logically possible world", it isn't logically possible that water isn't H20.

On the other hand, we can perfectly well imagine having experiences that would convince us (and that would make it rational to believe that) water isn't H20.  In that sense, it is conceivable that water isn't H20.  It is conceivable but it isn't logically possible!  Conceivability is no proof of logical possibility.

It appears to me that "water" is being used in two different senses in these two paragraphs—one in which the word "water" refers to what we type into the Science Interpreter, and one in which "water" refers to what we get out of the Science Interpreter when we type "water" into it.  In the first paragraph, Hilary seems to be saying that after we do some experiments and find out that water is H20, water becomes automatically redefined to mean H20.  But you could coherently hold a different position about whether the word "water" now means "H20" or "whatever is really in that bottle next to me", so long as you use your terms consistently.

I believe the above has already been said as well?  Anyway...

It is quite possible for there to be only one thing out-there-in-the-world, but for it to take on sufficiently different forms, and for you yourself to be sufficiently ignorant of the reduction, that it feels like living in a world containing two entirely different things.  Knowledge concerning these two different phenomena may taught in two different classes, and studied by two different academic fields, located in two different buildings of your university.

You've got to put yourself quite a ways back, into a historically realistic frame of mind, to remember how different heat and motion once seemed.  Though, depending on how much you know today, it may not be as hard as all that, if you can look past the pressure of conventionality (that is, "heat is motion" is an un-weird belief, "heat is not motion" is a weird belief).  I mean, suppose that tomorrow the physicists stepped forward and said, "Our popularizations of science have always contained one lie.  Actually, heat has nothing to do with motion."  Could you prove they were wrong?

Saying "Maybe heat and motion are the same thing!" is easy.  The difficult part is explaining how.  It takes a great deal of detailed knowledge to get yourself to the point where you can no longer conceive of a world in which the two phenomena go separate ways.  Reduction isn't cheap, and that's why it buys so much.

Or maybe you could say:  "Reductionism is easy, reduction is hard."  But it does kinda help to be a reductionist, I think, when it comes time to go looking for a reduction.

34

70 comments, sorted by Highlighting new comments since Today at 6:43 AM
New Comment

Consider two identical flywheels made of iron:

a) starts in room temperature, you apply force and make it spin. b) stands still but you heat it uniformily with several flames.

Suppose that in both cases the same amount of energy has been put into the flywheels.

In both cases the atoms are moving in high speed. Now if you look at the flywheels with an infrared camera would they look the same? This is not a rethorical question.

-1[anonymous]10yNon-rhetorical answer: No. You may now begin speaking rhetorically.

Huh, nobody's answered this.

The torched iron would emit more infrared than the spinning iron.

The reason is because thermal motion isn't just any ol' motion - it's motion that has had time to come to equilibrium between alllll the different ways the atoms in the solid can move. For example, the first atom could move left, and the second atom move right, and the third atom move left, and so on. All told there are as many ways for the atoms to move as there are atoms in the solid, which is more than 10^23, which is way more than the measly 1 way of moving that is "all atoms go around the center." In order to emit infrared light you need the atoms oscillating against their neighbors at high frequency, which is a big chunk of those 10^23 ways the atoms can move, but doesn't have anything to do with "all atoms go around the center."

"the conflation of conceptual possibility with logical possibility. To Sadi Carnot, it is conceivable that there could be another world where heat and motion are not associated. To Richard Feynman, armed with specific knowledge of how to derive equations about heat from equations about motion, this idea is [...] so wildly inconsistent as to make one's head explode."

Shouldn't we distinguish between logical and physical possibility? A world in which heat and motion are disassociated seems to be impossible in a rather different way than worlds in which the law of noncontradiction is false or 7,497 is prime are impossible.

Davis, that depends on whether you take the word "heat" to refer to "whatever actually melts metal" or "microscopic movements and collisions of molecules". See the various discussions of Hilary Putnam's "Twin World" dilemma for more along these lines.

If we look at something with the naked eye and see "plastic" and then look at it again with a super-microscope and see fundamental particles whizzing around, why does the second observation disprove the first or somehow make it an illusion?

Fact 1. This object, when looked at through the naked eye, looks like X. Fact 2. This object, when looked at through the microscope, looks like Y.

Even after you know Fact 2, Fact 1 is still true. Microscopes don't make liars of our eyes. I think the error is in not accepting human limitations to being with.

We a... (read more)

I'm still waiting for someone to tell me where the color is, in a universe made of colorless elementary particles.

The color is in your mind. Your brain creates the sensation of various colors based on the frequency of light coming into your eyes.

1wizzwizz47moTechnically, certain photons have colours. Our "colour" categories are excessively broad, but all photons can be described as having a particular colour, or being colourless.

Heat has to do more with equilibrium than kinetics.

Mitchell, clearly you haven't looked at http://en.wikipedia.org/wiki/Quarks#Color.

Seriously though, taboo 'colour' from your question. Exactly which bit are you still waiting to have explained?

Ben Jones said:

Seriously though, taboo 'colour' from your question. Exactly which bit are you still waiting to have explained?

I think you're tabooing overzealously. If I have a bunch of red balls and a bunch of blue balls, and ask 5 people to partition the balls into two subsets based on color, I would expect the same result from all 5 people. How do they do it?

I can't speak for the other 4 people, but I was taught to perform that trick in kindergarten. They even asked me which color was my favorite. Later, in high school, I learned about the difference between red and blue on the electromagnetic spectrum. And still later, in college, I learned about color perception in Psych and CompSci classes. For a while, I imagined that everyone had learned the same color facts that I had.

Then I ventured into a philosophy classroom, was exposed to G.E. Moore talking about "yellow" and "qualia", and I came to understand that not everyone had the same educational advantages I had.

1DanielLC9yThe philophical question of qualia has nothing to do with how the sensation of color gets to your brain. It's about what it does in your brain that makes you conscious of it.
-2bigjeff510yIn order to answer the question you need to first understand what color actually is. Once you do, the answer to both your and Mitchell's questions are immediately obvious.
2Mitchell_Porter10ySo what is it?
-6bigjeff510y

I would rather say that the observable consequences of the heat like nature of the universe are already included in the observable consequences of the kinematic like nature of the universe, so heat is redundent in this sence, though still a useful idea.

Relying on the potential existence of an Idealized Omniscient Science Interpreter feels a bit too much like divine revelation for my taste. The difference is rather then saying "Aha! This is what has actually been happening all along." I would say "Aha! This more accurately fits my observations."

Of course it's possible to have heat that's unrelated to molecular motion. Just consider frozen mustard or red peppers.

Question: How much of today's psychology will look to future scientists like attempts to measure the hotness of jalapeno peppers by thermometers?

3bigjeff510yThat's not heat, that's pain.

The heat analogy to consciousness is nothing new.

Chalmers explains and responds here.

Yes, this is old hat. See also my post on Misusing Kripke/Putnam, which explicitly explains why the analogy to 'water = H2O' (and similar a posteriori identities, like heat = molecular motion) is no help to the physicalist here.

Peter, your question doesn't seem to be the right one for illustrating your concern. The qualitative experience of color isn't necessary for explaining how someone can partition colored balls. Ignoring the qualitative experience, these people are going through some process of detecting differences in the reflective properties of the balls (which they subjectively experience as having different colors). We could create a reductive explanation of how the eye detects reflected light, how the brain categorizes reflective intensities into concepts like "br... (read more)

[tangent] Hi Brandon, you may find my post on The Problem of Other Minds to be of interest -- note that the usual justification is to argue inductively from analogy (others are externally similar to ourselves, so most likely have similar inner lives).

I think you're right that the diverse experience hypothesis (my red is your yellow, etc.) is 'illogical', at least in the weak sense of ad hoc or less than perfectly coherent/reasonable. It is logically possible, mind you -- there's no reason the would couldn't have turned out that way, if the laws of nature h... (read more)

I used to lead with the rallying cry of "conceivability is not logical possibility" but after some reflection came to the conclusion that logical possibility is just an arbitrary subset of conceivability and the entire framework of "logical possibility" should be rejected outright. "This is logically possible" carries no weight. It is not a thing. It serves no purpose and has no known use. By rejecting it what exactly do we stand to lose?

I interpret "$SITUATION is logically possible" to mean "$SITUATION can be modeled without any necessary contradictions"; thus, it's logically possible for energy not to be conserved (if we allow to change whatever laws need to change), but not for 21 to be a prime number (within any system of arithmetic like the ones we're used to, but then, if we used another system the relevant object/predicate wouldn't be what we mean by "21"/"prime") - or, more significantly, only one of P=NP and P≠NP is logically possible, even though with my knowledge I can conceive of both.

There's a way you could make the heat=motion concept much clearer to Carnot. When one studies kinematics, one generally makes the approximation that macroscopic bodies are rigid, and the motions of the body refer to center of mass motion, or perhaps rotation about some axis. If you explain that "heat" refers to the motion of the constituent particles relative to each other, I think a scientist of Carnot's day would understand the idea pretty quickly.

I think this sort of thing might be what people mean when they talk about a "bridging theory".

More significantly, only one of P=NP and P≠NP is logically possible, even though with my knowledge I can conceive of both.

Couldn't P=NP and P≠NP both be consistent with the standard axioms of complexity theory (whatever they happen to be right now), in the same way that, say, the different parallel postulates are all consistent with the rest of the commonly accepted axioms of geometry, or the way that the continuum hypothesis is independent of Zermelo–Fraenkel set theory with the axiom of choice?

Richard: Chalmers' argument on the example of heat and his claim that Dennet's conception of a vitalist is imaginary both appear to me to be correct responses to this style of attack on the "hard problem of consciousness".

Unfortunately, the post "Misusing Kripke" seems to me to be horribly confused. You assert that "Nobody thinks that the 'Twin Earth' world Putnam describes is an impossible one. Rather, we still grant the possibility of the world itself, but merely re-assess how best to describe it." and that "we can i... (read more)

Oh. Yeah. There is that option.

Just because I can conceive of a world in which what I experience as red is in fact experienced by someone else with no neural/optical flaws as what I would call yellow

What 'experience'? What properties does this experience have? Let me guess - you can't tell us.

How do you justify your assertion that those experiences exist? How would you convince an honest skeptic that there is more to your perception of mixes of different wavelengths of light than can be explained by referring to the properties of the eye, the signals the eye sends, and the neurology that processes those signals?

Richard:

In an earlier post of yours I find. "I'd add that mostly everyone accepts that the laws of nature are contingent, and that there could have been a law according to which lead would transmute into gold. So I'm presupposing that common background."

This is definitely not what I assume. Rather, what I assume is that the laws of nature are, from our perspective, the computations that give our experience as their output. An emergent level of description of these computations is as being "about" various fundamental particles describ... (read more)

Thinking Physics is Gedanken Physics - I loved that book, tore through it at some point while I was in high school. I never found anything else quite like it.

To those who ask where colour arises from in a colourless world; Wrong Question and Mind Projection bonus!

Imagine an alien civilisation that has, say, fourteen colours. Calling two adjacent ones by the same name would be as ridiculous to them as someone here calling green and yellow the same thing. Still want to claim that 'red' is part of the territory? There are wavelengths, and there is the human faculty to tell them apart at sufficient intervals. Anything else is map only. There is no red.

Say you taste an apple. You know that the sensation you are expe... (read more)

1bigjeff510yOn this point I like to think about gorgeous photographs of nebulae, which have been taken with infra-red telescopes. How can I see them if they are infra-red? The photographers simply adjust the wavelengths into the range that I can perceive, so I get to see the beauty of the nebula as though I could see into the infrared spectrum. The fact that you can slide red into yellow into blue just by adjusting the wavelength of light should make it immediately apparent that there is no substantive difference between red, yellow, or blue - only very slight differences in wavelength. We have evolved three types of cones that each respond to a specific slice of the electromagnetic spectrum (it is quite narrow for each), and these are represented as entirely different things in our mind for the purpose of distinguishing between them. So long as your cones and my cones are working equally well, and our visual cortex is functioning equally well, it's almost a certainty that blue, yellow, and red look exactly the same in my mind as they do in yours. This is because the electrical and chemical signals will be identical, and they will be processed in the same manner for each of us. It would be a great surprise to discover they "seemed" different. Many birds, it's worth noting, have a fourth type of cone that captures a slice of EM radiation in the ultraviolet, because most flowers reflect into the UV spectrum, while most other things do not. This makes distinguishing flowers from non-flowers significantly easier for the birds who feed on them (and the flowers who rely on the birds for pollination).

BTW, Zombie Michael, maybe you can help us. Do you feel conscious? If so, can you tell us whether you're actually conscious or just identical in absolutely every way to someone who's conscious, but lacking that super-special consciousness-juju?

Concession: vision is harder to think about than most sensations, because it seems so tightly bound to reality. Easy to think that what's on the back of your retina is the world.
Actually, since we rely upon vision so heavily, we know the most about the 'illusions' it can produce.

No one who has even casually studied how vision works can take "seeing is believing" seriously.

Imagine an alien civilisation that has, say, fourteen colours. Calling two adjacent ones by the same name would be as ridiculous to them as someone here calling green and yellow the same thing.

I don't think you need alien civilizations for this. Not all human languages have color words that map 1:1 to English color words. (I seem to recall that the word for "red" in Korean includes what English speakers would call "copper". I could be mistaken.)

1DSimon10yI also recall hearing that in Russian, there are separate words for "blue" and "light blue", just as English has a special word "pink" for "light red".
2moshez9yDunno about Russian, but Hebrew has them for sure -- "T'khelet" means "Light blue", "Kakhol" means "blue". I know quite a few bilingual ~5yo kids, who, if they're wearing a light blue T-shirt, will scream at you if you say "you have a Kakhol T-shirt" in Hebrew, but will happily agree they are wearing a "blue" T-shirt -- thus showing that sufficient lack of reflectivity can have two conflicting vision systems in the same individual. (BTW -- "light blue" is just an approximation, it's a specific shade of light blue).
1[anonymous]9yItalian has azzurro for ‘light blue’, but nowadays it's not that unusual to use blu for that colour when the difference doesn't matter. (I was thinking that this might be a relatively recent influence of English blue, then I remembered about a famous mid-20th-century song which refers to the sky as blu.) I don't think that necessarily follows.
0Alejandro19yIn Spanish there also a word for light blue, "celeste" (etymologically related to "sky"), and how appropriate it is to say just "azul" (blue) instead of "celeste" depends on context. For example, an Argentine would never call their flag "azul y blanca" (blue and white) since they are taught from childhood that it is "celeste y blanca". But they wouldn't mind saying the sky on a clear day is either "azul" or "celeste". A same piece of light blue cloth might be called "celeste" in some occasions, and "azul" in others--for example if one just wants to contrast it with a red one.
0[anonymous]9yYeah, in flags, football jerseys, etc. Italian is more precise too, using blu only for royal blue or darker, celeste for sky blue or lighter, and azzurro for intermediate shades of blue. (I think I saw a list of English--Italian ‘false friends’ which listed blu/blue as false friends, saying that It. blu corresponds to Engl. navy blue and Engl. blue corresponds to It. azzurro! In some specific contexts that might well be/have been the case, but I don't think there are that many people left who would normally use azzurro for (say) Uno [http://en.wikipedia.org/wiki/Uno_%28card_game%29] cards, let alone people who would consider it exceedingly weird if you called such a card blu.)
0[anonymous]9yYup! In Russian, dark blue is "golubuy" and light blue is "siniy".
0Prismattic8yThat's backwards. Apologies if this response is too late for you to care.
4wedrifid8yIf only English had words like azure, sapphire, smalt, aqua, turquoise, periwinkle, iris, cerulean, ultramarine, verdigris, waterspout, zaffre or cyan to distinguish such nuances of colour! Of course, the precise boundaries between the various areas of RGB that are given a label and commonly emphasised in speech will vary dramatically between cultures.
6Prismattic8yThere was an experiment done where participants were shown two shades of blue and asked if they were both the same, or slightly different. When the two shades of blue fell on different sides of the goluboy/siniy divide, Russian-speakers were much better than English speakers at distinguishing them, but they were no better when both would be goluboy or both siniy. Language distinctions do have cognitive consequences.
2Jayson_Virissimo8ySpeaking Russian is far from the only relevant difference between Russian-speakers and non-Russian-speakers. What experiment are you referring to, specifically?
3Prismattic8yThis one [http://www.pnas.org/content/104/19/7780.full].
3Jayson_Virissimo8yThank you.
1KnaveOfAllTrades7yStandard causal reversal: In Soviet Russia, biological colour distinction causes linguistic colour distinction!
6Alicorn8yThe difference is that, in English, "cyan", like your other listed examples except the ones that I think are actually green or purple, is a kind of blue (like sepia is a kind of brown, canary is a kind of yellow, etc.), while "pink" is not a kind of red. In Russian, golubuy is not a kind of siniy, and vice versa. I think color words in languages are really interesting. Some languages have only three very basic ones that aren't kinds of anything else, and these tend to be translated as "black", "white", and "red" - dark, light, and bright colors. (They translate the bright color as "red" because if you have lots of objects in the room and ask someone who speaks this language to point to the best example of that color, they'll pick something bright red.) These three words are privileged in English too: they're the only ones we modify with -en to indicate that something is becoming more that color (redden, blacken, whiten - never bluen or greenen).
2wedrifid8y"Cyan" is one of the least "blue" on the list and I almost omitted it. These days I'd even tend to consider some [http://en.wikipedia.org/wiki/Aqua_(color\]) of those others "shades of cyan". Too much study of web design tends to corrupt perception the same way native language does. Certainly. Words for other common concepts operate the same way, framing the way we think and influencing the way we slice up reality. But with colors the differences are far easier to study. What cannot be seen with a glance we can quantify with a digital camera.
1Alicorn8yI think it's reasonably likely that the general population of English speakers will consider cyan its own thing after a while, in large part for the reason you mention.
2wedrifid8yProbably, getting included in the 16 colour palette computers used at times and being the name for one of the subtractive primary colours brings it up a bit in popularity.
0linkhyrule57yHow often are most of those used? I recognize most, but I've probably used... five of them in the past year, and I thought verdigris was green. -_-
1wedrifid7yAbout half of them are used enough to be considered 'words' not 'scrabble words' according to my ad hoc classification scheme.
4Dojan9yJust consider this [http://blog.xkcd.com/2010/05/03/color-survey-results/]...
1Dojan7yOr the Mantis Shrimp [http://arthropoda.files.wordpress.com/2010/03/human-vs-mantis.jpg]...

Doesn't that make writing fantasy very hard? Wouldn't it automatically turn into Hard Sci-Fi?

1bigjeff510yYou should see how he does Harry Potter! (I think it's better than the original, every night I hope for a new chapter, this is practically torture!)

I don't mean to be pedantic, but why do you write "H20" instead of "H2O?"

4Lachouette7yI've searched all those comments because I couldn't believe I'm the only one bothered by that. The difference is actually hardly trivial; "H20" suggests there is a molecule made from 20 connected hydrogen atoms! The missing subscript doesn't help (isn't there a way to make subscript the number?). Of course I know it wasn't deliberate, and I guess no reader would interpret is as anything else but "water". For the sake of completion I would appreciate an edit, though.
2MalcolmOcean7yFwiw, all html or formatting aside, there's a fairly well-supported unicode subscript 2 [http://www.fileformat.info/info/unicode/char/2082/index.htm]. Water has the chemical formula H₂O. Sugar is C₆H₁₂O₆. See them all here [http://www.fileformat.info/info/unicode/block/superscripts_and_subscripts/list.htm]
1MugaSofer7yAhhh now I've seen it I can't unsee it.

For me this article is really actual since I've caught myself few times already thinking about pretty this relation between heat and motion. I am literate and educated person, I've even been learning math BUT nowadays I am trying to rethink all these stuff which is just packed in my head under labels similar to "heat vs motion". So actually it is not that hard for me in 2015 to think that heat and motion are probably different concepts :) I just know that I need to really think a lot and re-learn the basics in a proper way, not just re-reading bo... (read more)