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Math, Science, Knowledge, and Nature
Rick Garlikov

I want to ask two questions, each of which requires a little explanation or introduction to show the force of the problems at issue:

1) There are many sports where an object, such as a ball, is thrown in order to hit, or go through, a target. In some cases, such as football, the target is often a moving target. In basketball, though the basket itself does not move, the shooter is sometimes in motion relative to the basket when shooting, and players who are being passed to may be in motion when the pass is thrown. All these things involve trajectories and intersecting motion paths -- things which can be figured out by math and physics, using various equations and formulas. Yet athletes rarely know the math and physics that would be involved, and even if they did, there is not time to make the kinds of calculations that would be required, and that is not how anyone aims baseballs, basketballs, footballs, golf balls, tennis balls, or any other object one throws or hits through the air to hit a particular target. Just as an example, to show this is not as easy as one might think, a pitcher throwing a baseball 85 mph to a catcher who is about 60 feet away, is throwing the ball nearly 125 feet per second, and it takes just under half a second to reach the plate. In a half second, any object that is dropped or thrown on earth will fall more than 3 and 3/4 feet. That means a pitcher has to allow for a nearly four foot drop. If you were to shoot a ball at 85 mph out of a gun straight at the catcher's mitt, it would hit nearly 3 and 3/4 feet below it. Yet pitchers feel as though they are aiming straight at the glove. Physics says they must be actually releasing the ball at an angle that will take it up nearly two feet above the glove and then back down that same almost two feet in order to hit the glove. But that is not what pitchers, or anyone who throws a ball hard and "straight" feels like s/he is doing. How then do people learn the proper trajectories, force to apply, etc?

2) Surely when a ball goes into the air, the ball is not calculating the trajectory it needs to follow; nor is it likely that the earth is doing any sort of calculation either. Why then does the ball follow a path that can be calculated by humans or by computers, given the initial force and angle conditions? What is the relationship between the laws of math and science on the one hand and the way things actually work on the other? Why do things follow the laws of physics, for example? Or, to put it more accurately, why do the laws of physics work to let us know how to calculate trajectories in those cases where we do calculate them? And since the laws and observations and formulas of science and math involve not just trajectories and paths of motion, but all action in the universe, the general question is "does the universe work by the laws of physics?" Does the universe "follow" or "obey" the laws of physics? If not, why do these calculations work?

I want to propose, and then try to explain and support, answers to both questions. What follows is somewhat theoretical and tentative. I offer it for your consideration.

First, it seems to me that these questions are not very different from asking about the relationship of the rules of grammar to language and language learning, or the relationship between things such as the principles of art or music theory and art and music or enjoying/creating art and music. 

I also think that learning to read and learning to write and sight-read music (vocally) are similar sorts of phenomena. And I think the same kind of answers will apply to all these things.

Second, it seems to me that we learn to throw balls and other objects, and we learn many other sorts of things, such as language, by trial and error, using memory to keep up with the various trials and errors and the trials with correct results. We also are able to extrapolate to some extent when the variables are slightly different from what we are used to -- as with throwing a ball of slightly different weight or to a slightly different angle or distance from us, from what we may have practiced. This seems to me to take a tremendous amount of memory -or something like memory, in case we want to distinguish event memory or the memory of consciously learned things from this sort of "memory"-- that is quickly retrievable on demand as needed. 

The memory may reside in the brain, or it may reside in some way in or near the muscles and joints used to perform the tasks. I have memorized a few piano pieces a note at a time, and I cannot play at all by ear, nor can I read music with any speed at all. Learning a piano piece by memory is a painstaking task, and I have not done it very often. However, once I have learned a piece, being able to play it seems to be "in my hands", not in my brain. I do not necessarily mean that literally but am trying to express how it feels to me, all of which stems from, or is included in the following. I cannot think about the notes when I play them. I have to relax and let my hands just do the work. If I think about what I am to play next or what notes I am trying to find on the piano, I cannot play at all. It feels to me as if my hands know the patterns and where to go as I hear the music, not my brain. If a piano key is out of tune, it can throw off my playing the piece altogether; if the piano keys are wider or narrower than what I am used to, my hands will miss the right keys and I will not be able to play until I can practice on that piano and get used to the spacing of the keys and which keys are the ones I am trying to hit. If I get lost in a piece, the sheet music that I learned it from will not help me, except very slowly in the same way it helped me learn the piece in the first place; I cannot even find the place where I have become lost on the music simply by looking at the page. My hands seem to have to follow some patterns of motion and sound as I play, and anything that changes, in the sounds or the hand positions makes me unable to play. For me, playing the piano is not primarily, if at all, an exercise of conscious knowledge.

Learning to ride a bicycle seems not dissimilar. It took me a long time to be able to learn to ride, and it took two friends helping support me on the bike as I pedaled until I learned to have "the feel" of what it took to balance. I have taught many kids to ride bikes and their experience seems to be like mine -- there is no difference you can really consciously account for in what you are doing between when you couldn't ride and when you could. When you can't ride, you don't know what you are supposed to be doing to be able to ride; but once you are able to ride, you don't know what you were doing or not doing before that made you unable to ride. You have simply acquired the "feel" of what it takes to balance the bike. Somewhere your brain or mind or body has acquired the "memory" of the necessary positions and adjustments of how to balance the bike.

When many children throw a small ball for the first time or two "to you", they throw it down toward the floor part way between you and them. But soon they start to elevate their throw and get the ball to you (or over you). I think they are adjusting, without realizing it, to the effects of gravity on the ball after it has left their hand. If where they aimed it initially made the ball go too low, they start aiming higher until they get to a place where the ball goes where they intend it to go. That adjustment then becomes what the feeling of "aiming at that target" becomes. So by the time a baseball pitcher reaches high school, college, or the major leagues, he feels as though he is throwing straight at the catcher's mitt even though his body is making the adjustments for gravity and time and force that it has learned through trial and error at first and then trial and successful practice for the ensuing years.

Similarly, when children learn to speak a native language, they learn to pronounce words through emulation and through a process of trial and error and correction, so that when one first says, as in English, "I hitted the ball" and the correction comes back "You 'hit' the ball, not 'hitted' it" eventually s/he knows that the correct past tense of "hit" is also "hit." But the child has not learned this by understanding the rules and exceptions of regular and irregular verbs. Moreover when a child develops a particular regional accent, of, say, its parents and neighbors, it does so without conscious effort or knowing any rules for how to pronounce words. As an adult who is not a linguist, I don't even know whether there are any rules or symbols or patterns that describe dialect or accent other than phonetic spelling of words that add or drop a letter (as in a Bostonian "My idea-r is you should pahk the cah heah and we will walk the rest of the way"). But whether there are or not, that is not how children learn a native language or develop a particular dialect or accent. 

I once had a group of fifth graders memorize and individually recite some passages into a video camera and recorder. Then I played it back for them and they and I critiqued their "performance". None of them spoke with any expression in their voice, and many of them slurred words and phrases together in something not too far distant from a mumble, though one could make out what they were saying if one already knew the passages. Except the last student, who was a girl that spoke so unclearly that I did not critique her recitation because it seemed obvious to me that she had some sort of severe speech problem that was probably physical in its cause or nature. I didn't want to embarrass her, and I didn't know what to say. She was impossible to understand and she had no expression whatsoever. She had watched and heard everyone else and our critiques of their performances, and she watched her own performance. The next day she came in and asked whether she could recite into the camera a different passage that she had partially memorized, but that she would feel more comfortable partially reading from the book where she had found it. I wasn't going to deny her this, though it seemed a pointless exercise in pain. The difference was miraculous; she not only spoke clearly and with amazing expression, she made facial expressions into the camera that were perfect for what she was saying. A virtual Sarah Bernhardt. Apparently she had the skill to speak clearly and expressively but, until she had seen her "errors" on the video, did not know she was not doing so. She was able to self-adjust within 24 hours. That took in part a conscious effort to speak more clearly, but not likely a conscious ability to be able to do so, because it is unlikely that she could have learned to speak clearly and expressively overnight if she had not previously developed the ability to do so. The feedback was an important part of this change, but not the whole of it.

A trial and error approach also requires feedback about what the error is. Coaching is often necessary for that. Otherwise practice and trial and error will sometimes only reinforce errors or bad habits, or you may not happen across the trial that works. For example, I never could run and dribble a basketball that I could keep under control, particularly if I was not watching where the ball went as I bounced it. There are some tricks for developing the feel for that, if one cannot learn the skill oneself, but I never figured out the tricks or the technique and did not know it until I read about it while I was in my 20's. I could then see the technique would work, but I did not bother to develop it by then. I may have been too old or to uninspired to develop it by then. There are also techniques for shooting foul shots well or for shooting shots from the side of the basket, where you don't have a backboard to help judge the distance. Many of those things are difficult for some people to figure out for themselves. 

But some people have either a natural affinity for seeing how to adjust or don't need to make that much adjustment. There is film of Tiger Woods hitting golf balls as a young child under the age of five, and, to me, even more impressively, there is film of Andre Agassi as something like a three year old, hitting ground strokes -- and what is amazing was his timing, in that he hit every ball perfectly square no matter where it bounced near him. That is extremely difficult for most people to be able to do, but he was able to "judge" or just see where each bounce would go and he was able to coordinate his arm movement with that. 

What is impressive about Tiger Woods' swing as a child was that he already had the feel of what a "grooved" swing should be. That is difficult for most people. Most people trying to hit golf balls do all kinds of wrong things that they cannot feel they are doing, and that are very hard to help them change without really good teaching techniques. 

I do not think these abilities are that different in some important way from a child prodigy's being able to make music, which requires a coordination between the hand and the ear (or the ear "inside") rather than between the hands (and body) and the eye. Prodigies or talented athletes seem to require less error in their initial trials, and they seem to be able to recognize what adjustments need to be made and have the skill to be able to make those adjustments. The rest of us do not have that much recognition or "muscle memory" skills. I have never learned, nor been taught, to sight-read musical tone intervals. I presume it would require hearing musical notes and the pitch "distances" between them, while looking at those notes or intervals on sheet music. And I presume that for most people it would require some practice to be able to do that. I have learned to play music by sight with an instrument, but I have never been able to hear a tone in my head when I look at a note on paper. Nor can I judge what the difference between two pitches just by seeing the notes on paper. Many singers or musicians can though. My children's first piano teacher seemed particularly amazing to me in this way in that the way she learned a piano piece was to simply look at it until she got into her head what it should sound like; then she put away the music and just played on the piano what she "heard" in her head.

I also cannot sing. My sister, who can sing, and who hears pitch much better than I, thinks I cannot sing because I do not hear that I am "off" in pitch. I do hear that I am off. I know I sing terribly. My problem is that I do not know how to make my vocal cords make the correct sound. I can whistle in tune, however. And my sister cannot understand how I can whistle in tune without being able to sing in tune, since it takes the same hearing. She still thinks it is a matter merely of hearing. I see it as a matter of hearing and of being able to make an adjustment to what I am hearing -- a muscular adjustment to the error. I cannot do that with my voice. Just as I could not do that when I tried to teach myself the trumpet one summer. I could make certain notes, but I could not make other notes because I didn't know where they "were" on the trumpet, just as I do not know where notes are on a recorder or a flute. 

That is very different from my problem with speaking German with the proper accent or playing some violin pieces accurately. My college German teacher said my accent grated on her ears, and she would sometimes pronounce a word repeatedly to try to get me to emulate her pronunciation. It was hopeless because I thought I was already doing that. I could not hear how what she was saying was any different from what I was saying. If I play the violin, I may hit some notes just off enough that it bothers a musician a great deal, but that I cannot perceive -- at least when I am playing. Sometimes I can perceive errors when listening to myself on tape that I cannot perceive when I am listening to what I am playing as I play it.

I met a woman once who had moved to the U.S. from Russia. She had lived here for a year when I met her. She had come with her eight year old daughter, who was nine when I met them. When they came to the U.S., the woman already spoke English pretty well from having learned it in school, though she spoke with a strong Russian accent, and with numerous idiomatic errors. When they first came here, the daughter spoke absolutely no English. By the time I had met them, a year later, the mother talked as I described, but the daughter sounded exactly like any other American nine year old, perhaps even more articulate in English than most. She had absolutely no accent, and she could verbally articulate anything she wanted to. She often served as a translator for her mother in regard to idioms or anything that her mother could not say or understand in English. Moreover, she could, and sometimes did, mock her mother's English by speaking English with a heavy Russian accent, making idiomatic errors. It was amazing to see this.

Since then I have met a number of Hispanic children of immigrant parents, and what is interesting to me is that if they are under five or six years old, they can speak in fluent English and in fluent Spanish, but they do not seem to realize they are speaking two different languages. If you speak to them in Spanish, they answer in Spanish, and if you speak to them in English, they answer in English, just as a matter of course, but you cannot ask them, without perplexing them no end, how to say something in the other language. They do not seem to comprehend what you are asking them. And they do not seem to notice that you and their mother are speaking "differently" to them at all. Nor do they seem surprised that you or their mother might speak sometimes in English to them and sometimes in Spanish. They just simply answer in the language you spoke, whatever you said to them. If you ask bilingual adults who learned both languages in childhood how they did that, they tend to think they simply always did speak both or that they must have learned both at home (even though perhaps one of the languages was not spoken at all by their parents). It is difficult for many bilingual adults even to be able to answer what age they were when they realized they were speaking two different languages. One woman told me she thinks it was around age 6, but she wasn't sure.

What it takes to hit a ball or to throw a ball or to learn a native language or to play an instrument musically is, I think, at least the following things, whether these come naturally or through being taught: (1) practice (in some cases -- such as language, after exposure) that involves trial with cases of error and cases of accuracy, (2) the ability to discern any errors, and (3) their "direction" and (4) their "magnitude"; (5) the ability to adjust to correct the errors, (6) the ability to be able to instantly recall the proper muscle positioning once one has found and recognized it, and (7) the ability to extrapolate instantly to positions slightly different from the ones in one's muscle memory from previous practice.

Notice none of the above has to do with using calculus or physics or the knowledge of grammar or musical theory or the principles of acoustics. And the ability to do these things in one kind of endeavor does not necessitate any carry-over to others. If it did, athletes would be great musicians and actors, and vice versa. Moreover, none of this necessarily has anything to do with conscious memory or with learning things by studying them or trying to memorize them. Most people are graced with a statistically normal range of learning ability -- the ability to learn to walk, run, have a certain amount of coordination to play games or sports halfway decently, learn to speak and understand their native language, learn to read and write, do a modicum of typical mathematical calculations, etc. A few are unable to do those things and a few others are able to do them prodigiously well and/or to do other things prodigiously well that most of us cannot do very well at all, especially without good teaching or training.

But it seems to me that clearly people learn many of these kinds of skills, particularly athletic or kinesthetic skills, and hand/eye or hand/ear skills through trial and error and practice, and through some kind of memory of the different "feels" for how to do one's muscles (or vocal cords, and mouth, lip, and tongue positions, in singing and speaking) rather than through any math, physics, or other formal intellectual or theoretical means or principles. Learning tennis or basketball has virtually nothing to do with understanding the physics or mathematics of trajectories. And tennis is perhaps complicated by there being multiple minimal and maximal distances one can hit the ball at any given angle, depending on the force and height angle one directs the ball to clear the net but land inside the boundary lines. While a computer might be able to do the proper physics calculations in microseconds, humans cannot do them fast enough, and most humans could not do them at all within any amount of time.

The question then is what those theoretical principles and systems (physics, acrostics, music theory, grammar rules, mathematics, etc.) have to do with the actual workings of those things to which they can be successfully applied.

It seems to me that what happens is that people in some cases have discovered patterns in nature or in language or in music or art or any other invented activity and we are able to represent those patterns in symbolic, often mathematical, ways, and even to deduce in some cases logical consequences or bases for those patterns. 

Sometimes we make deductions through the manipulation of symbols instead of through the application of logic or non-symbolic reasoning(1) to knowledge about a phenomenon, but when we make deductions using just symbols, we cannot tell ahead of time whether they will necessarily apply to the phenomenon or not. For example, when you work trajectory problems algebraically, you sometimes get not only a real answer but also a "negative number" for something like time or height -- an answer which does not have any actual or real physical counterpart that it represents. And we tend to gloss over or ignore some of the numerical indiscrepancies we come across, or we refine our principles to take them into account. For example, although adding numbers of objects can be done through the addition of whole numbers, some objects cannot be added together this way. E.g., one water drop added to another does not necessarily give two drops of water but may give one larger drop. Adding a 35O F liquid or object to another 35O F liquid or object does not yield any 70O F mixtures or objects.

Even apart from math or other symbolic representations of phenomena, we get anomalies where the rules or patterns do not apply or where they give the incorrect deduction or answer. For example, in language, when you apply grammar rules strictly, you will often get erroneous or awkward constructions because the patterns are sometimes only approximations or have exceptions. Churchill gave the famous response to the petty war department clerk who would not process an urgent request for material needed at the front during World War II because the request ended with a preposition. Churchill is reported to have told or written the war department official "This is a situation up with which I will not put!" in order to get across the point that winning the war was more important than the grammar needed to do it, and that some incorrect grammatical constructions were actually less convoluted, clearer, and easier to understand, and should thus be allowed to stand as exceptions. Or we recognize exceptions for verbs that do not fit the normal tense patterns, and may even give them a group name, such as "irregular verbs" to designate that, and thus we have to correct a child who tries to form past tenses of such verbs in the regular way according to patterns he has noticed ("hitted", "drinked" or "drinked-ed", "touched-ed", etc.) -- meaning they do not fit the pattern or typical rule. The child probably is not following a rule for constructing past tenses, but is simply doing them according to a pattern s/he has, perhaps unconsciously, noticed.

The point about exceptions is important because when we come up with formulas, equations, and other symbolic ways of representing, describing, or portraying phenomena, what we do is to make the symbols fit the phenomena, so that we either note which cases they don't fit and say those cases are exceptions, or we find a way to make the exceptions also fit some kind of formula. For example, if the trajectory of an object in a vacuum was not exactly a parabola, we would have to find a formula or some way of representing the path it actually took. We would do that by plotting the path, say from high speed photographs or as it tore through a path of smoke or some other substance thought not to alter its path that much, and then we would try to figure out a formula that gave those same kinds of points. As long as trajectories with different angles, forces, or other variables proved to fit the same formula, we would then say we see a pattern and have the correct mathematical description of it. 

If trajectories in a gravitational field did not fit that pattern, particularly if no two trajectories seemed to act alike, we would not see a pattern (at least not an obvious one) and the formula for parabolas would not apply to trajectories in a gravitational field. Insofar as air pressure and resistance influence the trajectories, we probably do not have perfect parabolic paths and perfect compliance to formulas actually anyway. 

So, phenomena that display patterns fit formulas only because we have constructed the formulas to fit them and because we have not found exceptions to the patterns in those phenomena, and because if there were a few exceptions to otherwise consistent patterns, we would either designate them as such and in a sense say they don't count or that they don't discount the formulas "because" they are exceptions and the formulas do not apply to them, or because if there were exceptions to a simple formula, we might try to create a more complex formula that includes both the regularities and the irregularities. 

If you ask why phenomena fit patterns at all instead of just always being individualistic and different each time -- if you are asking why there are regularities in the universe at all -- there are a number of possible answers:

a) "Causation" may be the way it seems and patterns may exist because "like causes will produce like effects under like conditions". There may be some sort of underlying forces that make things happen in certain ways. And these forces may be real things, not just theoretical entities that are only shorthand descriptions for noticed patterns or correlations. (E.g., in physics, "force" is a shorthand term for the acceleration of a mass. We notice or posit forces when masses change their motion, but forces may not be "felt" or otherwise shown to exist.) Of course, we could still then ask, in a possible infinite regress, why those forces always act the same ways on the same kinds of objects. An example of a correlation that does not have the sort of causal factor it seems to is what happens in movies or electronic games, where, say, one object appears to hit another and causes a sound and some sort of motion. In "real life" we would say the noise and the motion were caused by the impact of the two objects, but in "reel life" the actual objects we see are a large number of individual different objects -- they are different pictures projected on a wall, one right after the other to give the illusion of impacts and continuing motion, etc.

b) We force a pattern or find some regularities even in the most irregular situations. That is, the patterns are man-made constructs, as in those cases where we might, for example, discover that if we convert passages in an Encyclopœdia Britannica article to numbers in certain ways, we find certain numerical patterns. It is not that the patterns were put there or that there is some underlying reason for their being there; it is that we happen to notice a pattern that may have occurred totally by accident. There are lots of possible patterns in even the most random situations, and thus finding such a pattern is more about our powers of discovery, imagination, and inventiveness than about how things occur or "have to" occur in nature. Had there been one more paragraph in the article in question, the pattern may not have been kept.

c) As above, we simply focus on patterns we have discovered or invented, and we ignore the many phenomena that seem to have no pattern, no rhyme or reason, and we then think there are patterns everywhere in nature. That is only because we are not noticing all the non-patterns, or the phenomena for which we have no pattern or "explanation".

d) In man-made phenomena, such as language, poetry, or music, patterns may emerge because they were unconsciously created at first because they sounded pleasant or interesting in some way. Perhaps there is some sort of combined physics and biology cause for the feelings of pleasure or interest they generate. 

Now because science is empirical, it deals with in certain ways only with surface phenomena, phenomena that can be observed, even when it is supposedly the underlying cause of some other phenomena. What is discovered is simply other patterns. Suppose a clever young person wants to figure out how a mechanical clock works and he opens it up and finds gear wheels and springs. He may even figure out why the different gear wheels were made the sizes they were, so that the ratios account for differences in seconds, minutes, hours. He may be able to understand why a certain size mainspring was used, so as to give a certain amount of tension to work the flywheel, etc. But no matter what he sees and understands, it will only be in relationship to his understanding of other springs and gears, and what he has learned about their patterns of interaction. He will not know why a spring exerts a certain amount of force or why gear wheel ratios form patterns of motion they do, or why solid things seem to push against each other even though there are spaces among their atoms that leave room to mesh with each other as liquid atoms often do when one liquid pushes against another.

If you take a radio apart to see what makes it work, you won't see anything but its parts, and its parts do not show why it works or how it works. I once followed instructions in a book and made a "crystal" type of radio out of a razor blade, some wire wrapped in a coil, a safety pin, a set of earphones I already had, and other wire that connected all these parts together and to a cold water pipe underneath a bathroom sink. From Dayton, Ohio, I was able to hear a station broadcasting out of Philadelphia, Pennsylvania. How those objects could allow that to happen is not in any way obvious from looking at them. When we see gears pushing against each other and moving them in certain ways, we only "see how" they work by seeing the pattern of what they do until that pattern becomes familiar enough to us to make it seem as though we see what they do. But we are only becoming accustomed to the pattern, not really seeing what is happening any more than we see what happens when liquids mix. We do not see why (some) liquids mix and why solids don't; and we in part distinguish between liquids and solids only on the basis of the behavior patterns we see them make. 

When we "take apart" atoms, we don't see what is inside; we only see the affects of whatever actions we have done that even make us feel we have somehow split the atom. What we find as we discover new things is new patterns, and what we invent when we invent things from such discoveries, is new ways to employ those patterns -- ways which happen to work when we are lucky or have deduced correctly, even if the deduction, though warranted in some way, is still only accidentally correct. There are plenty of warranted deductions that do not work. For example, it turns out that if you throw a raw egg high up in the air and let it land in grass, it will almost never break (on a single or first throw), as long as it lands in the lawn and not on a bare spot of dirt or against a stone or pebble. That is extremely counter-intuitive and no matter how often you try this (using a different egg each time), it will seem impossible that it won't break even though you have watched it not break each time in the past. The reason science has to be experimental and empirical is that we cannot simply tell what will happen from deductions. We have to test it to see whether expected patterns will hold up under new conditions. 

If they do not, we change the math or the formula until we find one that fits, but it fits because we make it fit, not because nature has done anything by that formula. As long as nature is consistent, there will be some formula or pattern or description of what is occurring, and the issue is one of our finding such a description. The patterns we discover are made to fit the phenomena. That does not mean the phenomena are made to fit the patterns or are made to follow the patterns. The patterns follow the phenomena, not vice versa.

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1. An example of a deduction by non-symbolic reasoning would be noticing that something which is otherwise out of reach might be reached by using a stick to essentially extend one's reach in order to retrieve it. An example of verbal logic is the proverbial "if all men are mortal, and you are a man, then you are mortal".
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