## Justin's Insight

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Justin (Robot) made headlines when it gained a new ability - catching objects thrown at it with an (?)80% success rate. The way the inventors made a big deal of Justin's catching ability is an indication of how difficult it is for a robot to copy even, what appears to us as, "simple" human motor skills.

I don't deny that a toddler will have a much lower success rate than Justin in catching thrown objects; this ability probably ranks higher in motor-skills ability than just grasping stationary objects. However, for the human child this ability will be learned and then mastered to a point that an 8 year old will probably beat Justin in a game of throw and catch.

I'm not particularly concerned about the relative difference between humans and robots in re the level of difficulty for this catching ability ; what I'm interested in is the difference between humans and robots in terms of how (the processes involved) the two catch balls.

When an object is thrown at me, and I hope I'm representative of the average human, I make an estimate of the trajectory of the object and its velocity and move my body and arm accordingly to catch that object. All this mental processing occurs without resorting to actual mathematical calculations of the relevant parameters that have a direct bearing on my success in catching thrown objects.

I tried looking up how Justin actually catches objects but it drew a blank on the www. However, I'm somewhat confident that Justin's catching ability involves actual mathematical calculations of a thrown object's trajectory and velocity and its own movements. The takeaway here being that without some actual number crunching Justin or, for that matter, any other robot won't have the ability to catch thrown objects.

So,

1. It's impossible for Justin to possess an ability to catch thrown objects without actually performing some mathematical calculations.

2. Humans possess the ability to catch thrown objects and we, unlike Justin, routinely catch objects without even thinking of mathematics let alone doing any actual calculations.

This is problematic to say the least. Justin suggests that we actually do perform mathematical calculations but are just unaware of it; it happens subconsciously so to speak. I'm not claiming this is the case but it seems to fit the data quite well: by data I mean what people refer to as insight. Insight has been described as a sudden realization of key features of a problem and generally leads to a solution to that problem. Insight is thought of as distinct from deliberate conscious thinking on a problem or issue and occurs when we're actually engaged in some other activity not related to the problem we gain insight on. It seems then that thinking, logical thinking and even complex mathematical calculations are being performed subconsciously in our minds. Thus I offer here what I think is an insight on what insight could be.

The other possibility is that we don't need mathematics to catch a ball and roboticists are barking up the wrong tree. Roboticists need to rethink their approach to the subject in a fundamental way. This seems, prima facie, like telling a philosopher that logic is no good. Preposterous! However, to deny this possibility is to ignore a very basic fact - humans don't do mathematical calculations when we play throw and catch, at least not consciously.

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I tried looking up how Justin actually catches objects but it drew a blank on the www. However, I'm somewhat confident that Justin's catching ability involves actual mathematical calculations of a thrown object's trajectory and velocity and its own movements.

I would not be too sure about that. For example, in the translation business, they moved away from algorythmic approaches to fuzzy learning. When Google Translate did that, the change almost over night was mind boggling. The service went from a joke to something that almost flawless in large sections.
• 4.9k
From Andy Clark's Being There:

"Consider the act of running to catch a ball. This is a skill which cricketers and baseball players routinely exhibit. How is it done? Common experience suggests that we see the ball in motion, anticipate its continuing trajectory, and run so as to be in a position to intercept it. In a sense this is correct. But the experience (the "phenomenology") can be misleading if one believes that we actively compute such trajectories. Recent research suggests that a more computationally efficient strategy is to simply run so that the acceleration of the tangent of elevation of gaze from fielder to ball is kept at zero. Do this and you will intercept the ball before it hits the ground.

Videotaped sequences of real-world ball interception suggest that humans do indeed — unconsciously — use this strategy. Such a strategy avoids many computational costs by isolating the minimal and most easily detectable parameters that can support the specific action of interception"

In other words: catching the ball isn't a matter of 'computation' so much as keeping constant a certain relation of movement between ball and person: an attempt to keep something invariant in your visual field.
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Justin suggests that we actually do perform mathematical calculations but are just unaware of it; it happens subconsciously so to speak.

Not at all. It's the requirement of computers - they process binary code, and anything they're programmed to do must be coded. But it's a way of modelling reality, not reality itself. Russell said 'Physics is mathematical, not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover. ' I would modify that to say that in order to model a process in a computer, it has to be quantifiable, but that organisms don't have to do that.

The impressive aspect of the 'Justin' experiment is that catching a ball is a 'fuzzy' problem - the ball can move in a variety of directions and a variety of velocities. That's what requires the smarts.
• 6.8k
Recent research suggests that a more computationally efficient strategy is to simply run so that the acceleration of the tangent of elevation of gaze from fielder to ball is kept at zero.

Right, if you're a sailor you'll know that avoiding the zero tangent is how to avoid a collision. And, if you're a storm chaser, and the tornado isn't moving to the right or the left, it's coming right at you. These are the situations when evasive maneuvers are called for.
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humans don't do mathematical calculations when we play throw and catch, at least not consciously.

Not doing one type of calculation consciously does not imply much about doing other calculations. Not being aware of doing one type of calculation implies even less. The mental model of calculations being done in the brain, then passing through consciousness, then the body reacting isn't particularly apt. It's more like perception data in our neural nets feeds forward to motor control functions and our internal models at the same time but in different ways. It feeds forward with most of the focus payed to immediate environmental differences, like changes in the visual field corresponding to the ball moving, and changes in the ball's relative trajectory relative to promoted motor functions.

There's no guarantee that even if a calculation of type X was somehow used by the body to interface the perceptual inputs with the motor outputs that it would feed forward in precisely the same way to our internal models, so we wouldn't necessarily become aware of its character even if it was happening as described.

There is also no guarantee that Justin uses the same interfacing strategy between its perceptual data and its motor functions that we do just because they have the same outputs; we'd be better off comparing Justin's behaviour in catching the ball to human behaviour catching the ball (say, with eye tracking goggles and a body sensor) to see if they do strategies with similar outputs (behavioural incentives and catching strategies).

I see that @StreetlightX made much the same point with a quote.
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Roboticists need to rethink their approach to the subject in a fundamental way.

And they did. The price of the neural network revolution was giving up (or at least severely compromising) the model of the brain (or computer) as a processor of stored symbols - internal words and pictures representing external objects. Ironically, it had to revert to Skinner's behaviourist model, a "black box". Training, without necessarily understanding the learning.

How Justin does it is as much up for speculation as how we do it.

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Noted!

Well, I
Recent research suggests that a more computationally efficient strategy is to simply run so that the acceleration of the tangent of elevation of gaze from fielder to ball is kept at zero. Do this and you will intercept the ball before it hits the ground.

Sounds very mathematical.

Imagine Justin and me in a room and objects are thrown at us and we manage to catch them with equal success rates. While we may differ in appearance we're practically identical in terms of ability, catching thrown objects. By analogy, hopefully not a poor one, the processes that confer the catching ability should also be similar if not identical. I wouldn't go so far as to say this must be so but it does seem a reasonable inference to make.

Also, catching is an ability that evinces precision: setting aside acceptable error margins because of the sizes of the object thrown and the size of our hands, the whole activity requires some level of exactitude in the velocity of the arm and the amount of force I exert on the thrown object; move even slightly slower/faster, exert an inappropriate amount of force, and catching can't be accomplished. I guess what I'm getting at is there's a level of precision in catching that points to some form of computation process going on in our heads, somewhere but concealed from our consciousness.

Russell said 'Physics is mathematical, not because we know so much about the physical world, but because we know so little

You always find the right quote for the right occasion. Thanks. I understand that the reason why math is so highly regarded is because it's the only game in town and there may be more to physics than just numbers. What intrigues me is that human abilities, albeit only the physical, can be reasonably mimicked by mathematical computational models.

Strip it down to the basics for clarity: it's obvious that a thrown object's position at the moment the catch is made is completely determined by its exact velocity (speed and direction). If I or a robot must intercept the ball, we must move our arm at the correct velocity and take measures to apply the correct amount of force on the object thrown at us. The point here being that there's a level of precision in the act of catching that, to me, eliminates the possibility that it's done using rough estimates like I was suggesting because the error margin involved is very small; even a small deviation from the correct velocity for our arm or the force we apply will result in a failure make a catch. Basically, catching throw object's isn't fuzzy. Ergo, it seems probable that our brains actually do the math when catching thrown objects and by extension, when doing other physical activities.

An interesting thought on this is that, in line with Russell whom you quoted, physics may have a non-mathematical dimension to it and our brains tap into it when we do something physical without the need to deal with actual numerical calculations. This option is almost too tempting to resist, for me at least, but all it takes for it to be falsified, if only for the issue at hand, is to examine what actually goes on in our minds when we're in the act of catching objects thrown at us: from personal experience I'm under the impression that we take note of the speed and direction (velocity) of the object thrown at us and move accordingly. So, yes, there may be, as Russell claimed, a non-mathematical side to physics but it doesn't seem to figure in the act of catching thrown objects.

I see that StreetlightX made much the same point with a quote.

:ok: :up:
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By analogy, hopefully not a poor one, the processes that confer the catching ability should also be similar if not identical.

Why would you ever conclude that? If a group of people show up at work in the morning do you conclude that they took similar, if not identical modes of transportation, just because they demonstrate that they have the capacity to get to work?
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Sounds very mathematical

Sorry, I should have realized I needed to translate my quote into dumb. Thing move, I move, must make it so that one thing move in certain way in relation to other thing; if move good, catch ball.
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I'm working on a fairly advanced AI project. It has quite a lot to say about the notions in this thread...

@StreetlightX

You're right to say that these kinds of problems are not necessarily solved via computation. The very structure of our bodies and the environment (directional gaze/perspective + 3D motion) make it so that we can just line up a few "values" (the horizontal displacement of the ball with respect to one's own change in velocity in this example) to intercept the ball...

However...

Much more trivial than learning how to catch a ball is learning to control one's limbs in the first place. Running involves coordination and sequence timing of hundreds of individual muscles, and running smoothly requires high-speed feedback reflexes and modulatory signals both from the external environment (like when your foot touches the substrate) and from different parts of the CNS.

Running is actually a much more complicated problem than just catching a ball (catching is a brief episode with very specific goals and requirements) because running requires very fine coordination of the entire body. Staying balanced, applying the rightforce ratios, timing our contractions, and responding to perturbations is the complexity that prevents us from creating good humanoid robots. Yes, Boston Dynamics spent ten years making a hard coded algorithm that can run on a very specific course that it has been optimized for, but it cannot *run in general* (meaning if you change the course slightly or introduce anomalies, it will fail).

A common way to describe this problem is to call it "the curse of dimensionality". Imagine that our muscles are controlled by a series of dials that determine contraction strength. There are about 640 knobs that each control individual muscles in the human body. Almost any random combination of settings on these knobs will result in seizure like behavior. The person would fall over and begin writhing randomly (and probably injure itself in the process). Because there are so many individual values, and because changing one of these dials can drastically change what happens to the body (example:walking would be a specific sequence of dial settings, but if you suddenly activate a random muscle while walking, it might cause the body/agent to fall over or fail), figuring out how to correctly manipulate these dials for good motion patterns is extraordinarily difficult. (the problem space is too large to search with regular machine learning algos, and it's also too complex to have the motion rules hard-coded by hand).

This brings me to the spine and "central pattern generators". These are neural circuits that hard-code primitive and basic motion patterns (from utero), and this essentially biases out a large portion of the problem space referred to earlier. Exmaple: most walking patterns are very similar, even across different species of animals.The fundamental base of the pattern can be more or less hard coded in a primitive way with these pattern generators, and this gives the new-born agent an obvious starting point when it is learning to walk. Ontop of this, these central pattern generators can be hooked up with autonomic reflexes that essentially solve part of the walking problem automatically (example: they can apply greater force when they receive a signal that one of the legs or feet are stuck,which can be based on rudimentary proprioception signals).

Chickens can run with their heads cut off (because the pattern generators in their spine are already wired and optimized for running) and baby deer can stand up almost instantly for the same reason. This is why when we ourselves want to walk or run, we don't have to think about each and every muscle, or each and every foot fall; we just think a high level thought "run", and our lower brain and spine basically sorts the rest out for us.

In other words, the non-intelligent body (the structure of the body, the way the reflexes are wired up, and the default spinal systems that simplify control for the CNS) is actually a fundamental part of higher intelligence. Relating this back to the ball-catching problem, the only reason that we're physically dexterous enough to even contemplate catching a ball is because the unconscious parts of our brain and body do most of the work for us.

Another ramification of this realization is that it's unreasonable to expect current neural network architecture to be able to express itself with broad/general intelligence when it is given a complex set of outputs to control.

How can an agent learn to speak if controlling the voice box is too complicated? How can it paint or play soccer if the problem of ambulating an arm and a leg is beyond the ability of our best networks?

How can an agent learn to communicate using language if it cannot learn and relate language-encoded concepts using the myriad of other senses that may be required for experiencing the concept or thing being described? For example, how will a language transformer ever understand what a cup actually is if it has no experience of 3d space? If it has never explored or touched a cup? Can it truly understand cups if it is too far away from a thing that could actually use one?

We see all kinds of wonderfully good results coming out of machine learning, but they're all stupendously narrow functions. We can train an agent to identify animals within images, or we can train it to balance a ball, but not both. We cannot yet integrate these different intelligence into a single AI with any degree of elegance whatsoever. We're still behind in this respect because we have yet failed to comprehend how animal intelligence is built from the bottom up in the first place. Philosophically inclined ML devs (especially) only seem to concern themselves with high level cognition where concepts and ideas are already there to be played with, having no sweet clue where they really come from beyond "it's the magic of end to end learning", and hence they get nowhere but to produce lofty theories about how it all might work (they're in for decades of speculation).
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Excellent post. But I've seen some of those spooky Boston Dynamics robot videos, and they're pretty darn good at freestyle running!

Ergo, it seems probable that our brains actually do the math when catching thrown objects and by extension, when doing other physical activities.

Nope. I'm sure maths has nothing to do with how organisms perform such actions. They can be modelled mathematically, but (for instance) a chameleon capturing an insect with its flexible tongue or an owl catching a field mouse through its acute sensory abilities requires great accuracy but absolutely no mathematical ability.

History - recall the breakthrough in science that arose from the combination of Rene Descartes' algebraic geometry (described here) combined with Newton's laws of motion. These were two of the foundations of what was then called 'the new science' - which indeed it was. Science since then has built on those foundations by using that methodology to model all kinds of processes - anything that moves through space can theoretically be modelled by such a methodology (which is why it was key to modern science generally). That's where the mathematical modelling of catching a ball comes in - it requires very sophisticated mathematics to allow for all the millions of possible variations. But when you and I catch a ball, we don't rely on mathematics at all. What mathematics does, is allow us to mathematically model such processes.
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Why would you ever conclude that? If a group of people show up at work in the morning do you conclude that they took similar, if not identical modes of transportation, just because they demonstrate that they have the capacity to get to work?

Your analogy is too weak because there are relevant dissimilarities like the differences in the distance of the employees' homes from work which would imply they would at least have to make their trip in different forms of transportation or at different times.

In my robot-human analogy, not only is the ability to catch near-identical when observed but the methods employed seem to be similar in terms of needing quantification (math). This isn't hard to prove; just observe yourself catching something in the air. If a ball is thrown at you, you'll notice yourself adjusting your arm's and body's motions to intercept the ball as per its speed and direction (velocity). The conscious determinants of these bodily adjustments are very vague i.e. what you'll be aware of maybe just very vague descriptions of the ball's movement such as it's very fast, fast, medium, slow, very slow, etc. As is obvious these are instances of quantification (math) albeit in very vague terms or so it seems. My contention is that there's a level of precision in catching thrown objects that require actual mathematical computation; after all, if you move your arm even a little faster/slower and the direction of movement is off by more than a few degrees, you'll fail to make the catch. Ergo, the vagueness in the necessary quantification/math involved in catching ability is not the real truth of the matter; the precision required entails actual math to be done.
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So, there's something non-mathematical in human and animal physics? If that's true then how come mathematical physics is applicable to kinesiology and biomechanics; after all human limbs are essentially mechanical levers and the amount of force muscles can exert can be quantified. I don't see how the body's physical abilities are quantifiable mathematically and yet the brain controls it non-mathematically. At the very least the applicability of physics, a mathematical enterprise, to our bodies indicates that somewhere along the chain from intending a movement to the actual movement itself there is some math involved.
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Excellent post. But I've seen some of those spooky Boston Dynamics robot videos, and they're pretty darn good at freestyle running!

Thanks!

Boston Dynamics has some quadruped robots that aren't too bad, but it's a much more stable and simplified body, and they're still quite fail-prone.

The humanoid robot simply cannot do freestyle running. All we ever get to see are the very best results on specific setups that they try over and over again. And to boot, they won't even talk about their underlying methods (cause it's embarrassing hard coded rules systems that is over-complicated, hard to generalize with, and they don't want a decade + of mind numbing effort to be stolen by competition)
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I don't see how the body's physical abilities are quantifiable mathematically and yet the brain controls it non-mathematically

I don't think it's that hard to see. Remember the mathematization process is a method - that's why I mentioned its history. I mean, before Descartes came along nobody had ever thought of modelling three dimensional space as geometrical co-ordinates. (This is one of the discoveries that qualifies Descartes for the title of genius.)

Almost anything physical is quantifiable using that method insofar as it has mass, velocity, and other attributes that can be quantified. That methodology was very much the consequence of the discoveries of Newton, Galileo and Descartes, among a few others - crucial to modern science. Nobody from before their time thought about things that way. And that methodology is universal in scope - you can use it to model almost anything from the atomic to galactic scales (with the caveat that the discovery of relativity and quantum theory have shown that Newtonian physics is only universal within certain scales.)

So, that methodology is what is used in robotics, artificial intelligence, and so on - it all relies on the computation of quantifiable attributes. It's not that there's something intrinsically mathematical about what's being modeled (in this case, although there might be in other subjects). It's simply that mathematical modelling is what is behind all such technologies.

The whole question of 'what is maths' and 'what is number' is also a really interesting one, but it's not actually connected with the question of how 'Justin' does its stuff.
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I don't think it's that hard to see. Remember the mathematization process is a method - that's why I mentioned its history. I mean, before Descartes came along nobody had ever thought of modelling three dimensional space as geometrical co-ordinates. (This is one of the discoveries that qualifies Descartes for the title of genius.)

Almost anything physical is quantifiable using that method insofar as it has mass, velocity, and other attributes that can be quantified. That methodology was very much the consequence of the discoveries of Newton, Galileo and Descartes, among a few others - crucial to modern science. Nobody from before their time thought about things that way. And that methodology is universal in scope - you can use it to model almost anything from the atomic to galactic scales (with the caveat that the discovery of relativity and quantum theory have shown that Newtonian physics is only universal within certain scales.)

So, that methodology is what is used in robotics, artificial intelligence, and so on - it all relies on the computation of quantifiable attributes. It's not that there's something intrinsically mathematical about what's being modeled (in this case, although there might be in other subjects). It's simply that mathematical modelling is what is behind all such technologies.

The whole question of 'what is maths' and 'what is number' is also a really interesting one, but it's not actually connected with the question of how 'Justin' does its stuff.

Well, to be fair, there is no reason why there shouldn't be non-mathematical determinants of motion. However, given that we can model motion based on only math it seems either these non-numerical determinants of motion are superfluous or operate in parallel to the mathematical ones. Do you have any idea what such a system of non-numerical determinants of motion that makes predictions possible would look like?
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Do you have any idea what such a system of non-numerical determinants of motion that makes predictions possible would look like?

I'm the first to agree that there are many things that can't be quantified, but I can't see how this is one of them.
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To all (if interested)

I think most of the replies in this thread referred to learning and I wish to build up on that to make the case that our brains actually do math with our bodies.

As we all know Justin can't learn i.e. his success rate with catching is never going to improve unless he gets a software upgrade. The nature of this upgrade will be faster computation ability in tandem with more accurate measurements of relevant parameters (mass, velocity, etc).

Humans and to a limited extent other lifeforms are, unlike robots, well-known for their learning ability. I've heard people say "practice makes perfect" and this maxim is nowhere more appropriate than physical activities, specifically sports. Practice is the key to becoming a great sportsman and although people speak of talent, it goes without saying that talent is somewhat secondary to practice. How does practice lead to an improvement of a physical ability?

Let's look at basketball as a sport because I have a personal story to tell that's relevant to the topic. In my opinion, what practice does is it helps improve our estimates about relevant parameters. Balls are standard and so there'll be minimal deviations of the relevant quantity, mass. So, the more we practice, the closer our measurement of the ball's mass to its actual mass and that allows us to exert the right amount of force to score a basket. In other words, practice is nothing more than getting an accurate measurement of a key parameter in a a sport.

I know we don't actually get a reading of the ball's mass like a weighing scale; it's more of a feel. Nevertheless, getting familiar with the game is another way of saying measure key parameter accurately.

Now my story: I remember playing basketball and although I'm not really good at it, I recall practising enough to see an improvement in my performance. Initially the ball wouldn't do the thing I wanted it to do but slowly I began to get accustomed, as it were, to the ball and it would then with less effort do as I intended. One day I decided to use a volleyball to play basketball and to my dismay I went back to being a complete novice. Now, the only noticeable difference here was the mass of the ball, volleyballs weighing less than basketballs, and mass is a determinant for how much force needs to be applied to the ball to make it follow a certain trajectory.

Ergo, given what practice is and how my story reveals that changing the key physical quantity involved in a sport can turn you from a pro to a beginner, it must be that our brains do math; after all, all that changed in my story was a mathematical quantity, the mass of the ball and it resulted in a difference in performance. It isn't much of a problem to stretch this conclusion to all physical activities.

Note to @Wayfarer: changes in a number (physical quantity) can cause a difference in performance. Doesn't that indicate math is involved? If math wasn't a part of it then tweaking with the numbers shouldn't have an effect on our physical performance.
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Textbook example of putting the cart before the horse.
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Textbook example of putting the cart before the horse.

How? Which is the cart and which is the horse?
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The reasoning is quite simple actually.

1. Either catching involves math or catching doesn't involve math

2. If catching doesn't involve math then changing mathematical parameters (like mass) shouldn't affect catching ability

3. Changing mathematical parameters (like mass) does affect catching ability

So,

4. It's false that catching doesn't involve math (2, 3 modus tollens)

Ergo

5. Catching involves math (1, 4 disjunctive syllogism)

6. No perceptible difference exists between catching and other physical activities i.e. they appear to be similar in process

Ergo

7. All physical abilities require the brain to do math
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1. Either catching involves math or catching doesn't involve math

2. If catching doesn't involve math then changing mathematical parameters (like mass) shouldn't affect catching ability

But you're not comparing like with like. When a human catches, then the action consists of muscular reflexes, hand-eye co-ordination, and on a micro-cellular level, the exchanges of ions across membranes, and so forth and so on.

Those actions can be modelled by machines, but that modelling relies on maths.

When a machine performs an action, then you have motors which position instruments controlled by binary code. This is reliant on mathematics, in a way which is completely different to the way in which organic performance is.

If you can't see that, I give up. (But then, this should have been obvious from the way the thread was named, as nothing about what 'Justin' can do, connotes 'insight', and indeed the word is not to be found in the linked Wikipedia page.)
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But you're not comparing like with like. When a human catches, then the action consists of muscular reflexes, hand-eye co-ordination, and on a micro-cellular level, the exchanges of ions across membranes, and so forth and so on.

Those actions can be modelled by machines, but that modelling relies on maths.

When a machine performs an action, then you have motors which position instruments controlled by binary code. This is reliant on mathematics, in a way which is completely different to the way in which organic performance is.

If you can't see that, I give up. (But then, this should have been obvious from the way the thread was named, as nothing about what 'Justin' can do, connotes 'insight', and indeed the word is not to be found in the linked Wikipedia page.)

I'm offering a very simple argument here. Compare physical abilities to that of the ability to discern color. If, as you say, the physics of human motion is non-mathematical then this comparable to saying we don't discern colors.

What would be a simple test to prove/disprove the two theories above, yours about our brains not doing math and the other that we don't discern colors?

A simple test would be to show us a variety of colors and check if different colors produce different responses and by analogy if we wish to check if math is an integral part of our physical abilities we should introduce variations in mathematical parameters we know are involved. If there is, and there is, a noticeable difference, in our response to different colors and in our physical performance, to these variations then it's quite clear that both we can discern color and our brain does math, right?
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n my robot-human analogy, not only is the ability to catch near-identical when observed but the methods employed seem to be similar in terms of needing quantification (math).

Your analogy is worse than mine. A human being doesn't use math to catch a ball. That's a false premise. You admit this yourself when you say a human beings adjustments are "vague". Math is not vague.

So, there's something non-mathematical in human and animal physics? If that's true then how come mathematical physics is applicable to kinesiology and biomechanics; after all human limbs are essentially mechanical levers and the amount of force muscles can exert can be quantified. I don't see how the body's physical abilities are quantifiable mathematically and yet the brain controls it non-mathematically. At the very least the applicability of physics, a mathematical enterprise, to our bodies indicates that somewhere along the chain from intending a movement to the actual movement itself there is some math involved.

The fact that mathematics cannot adequately predict human motions, because it cannot predict changes due to free will decisions, ought to indicate the falsity of that premise to you. Human motions cannot be modeled with mathematics.

Well, to be fair, there is no reason why there shouldn't be non-mathematical determinants of motion. However, given that we can model motion based on only math it seems either these non-numerical determinants of motion are superfluous or operate in parallel to the mathematical ones. Do you have any idea what such a system of non-numerical determinants of motion that makes predictions possible would look like?

Perhaps we model motion only with math, but math is inadequate for modeling human motions. So you ask what is the nature of such "non-numerical determinants of motion", and the answer is conscious judgements. A mathematical judgement is one very specialized type of conscious judgement. However, the vast majority of conscious judgements, including those which induce motion are not mathematical judgements.

To disprove your theory, just watch a dog jump for a stick, or a treat, then try to get the dog to apply some mathematics. I'm pretty sure that the dog catches without applying mathematics.
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Ergo, given what practice is and how my story reveals that changing the key physical quantity involved in a sport can turn you from a pro to a beginner, it must be that our brains do math

This doesn't follow at all. Just more idiotic reasoning, as with all of your posts.
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Well, by what means could I determine if the brain does math or not when involved in a physical activity? How do we know that our brain has the ability to recognize different colors?

Change a mathematical quantity and observe differences in the physical activity that involves that mathematical quantity. Change colors and look for changes in response.

If there's a noticeable difference in either the physical activity or the response to different colors then can I not conclude that our brain does math and that we can discern different colors?

In fact color is an excellent example of our brain doing math because we can discern colors and color is completely determined by a mathematical quantity viz. frequency of EM waves.
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This doesn't follow at all. Just more idiotic reasoning, as with all of your posts.

Noted. Will work on that. :up:
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1. It's impossible for Justin to possess an ability to catch thrown objects without actually performing some mathematical calculations.

2. Humans possess the ability to catch thrown objects and we, unlike Justin, routinely catch objects without even thinking of mathematics let alone doing any actual calculations.

You seem to be conflating "performing" with "thinking". Does Justin "think", or "perform"? Is there a difference between "performing" mathematical calculations as opposed to "thinking" of mathematical calculations?

Not at all. It's the requirement of computers - they process binary code, and anything they're programmed to do must be coded. But it's a way of modelling reality, not reality itself.
What about brains? Are brains programmed? The model is just as much part of reality as what is being modeled. The model has causal relationship with what is being modeled and has causal power itself (it changes your behavior based on the model and what is being modeled).

Thing move, I move, must make it so that one thing move in certain way in relation to other thing; if move good, catch ball.
Looks like you are being run by an IF-THEN program. A high-level language is a representation of the machine language that computers understand. So is your mind a representation of what is going on at the neurological level. You're not aware of the mathematical calculations your neurons are performing. You mind's mental imagery is a representation of what is going on at the neurological level, just as you aren't aware of what is going on inside the computer by just looking at the screen, but the screen is a representation of what is happening inside the computer.

Here is an excerpt from Steven Pinker's, How the Mind Works, that might shed some light here:
Mathematics is part of our birthright. One-week-old babies perk up when a scene changes from two to three items or vice versa. Infants in their first ten months notice how many items (up to four) are in a display, and it doesn't matter whether the items are homogeneous or heterogeneous, bunched together or spread out, dots or household objects, even
whether they are objects or sounds. According to recent experiments by the psychologist Karen Wynn, five-month-old infants even do simple arithmetic. They are shown Mickey Mouse, a screen covers him up, and a second Mickey is placed behind it. The babies expect to see two Mickeys when the screen falls and are surprised if it reveals only one. Other babies are shown two Mickeys and one is removed from behind the screen. These babies expect to see one Mickey and are surprised to find two. By eighteen months children know that numbers not only differ but fall into an order; for example, the children can be taught to choose the picture with fewer dots. Some of these abilities are found in, or can be taught to, some kinds of animals.

Can infants and animals really count? The question may sound absurd because these creatures have no words. But registering quantities does not depend on language. Imagine opening a faucet for one second every time you hear a drumbeat. The amount of water in the glass would represent the number of beats. The brain might have a similar mechanism, which would accumulate not water but neural pulses or the number of active neurons. Infants and many animals appear to be equipped with this simple kind of counter. It would have many potential selective advantages, which depend on the animal's niche. They range from estimating the rate of return of foraging in different patches to solving problems such as "Three bears went into the cave; two came out. Should I go in?"

Human adults use several mental representations of quantity. One is analogue—a sense of "how much"—which can be translated into mental images such as an image of a number line. But we also assign number words to quantities and use the words and the concepts to measure, to count more accurately, and to count, add, and subtract larger numbers. All cultures have words for numbers, though sometimes only "one," "two," and "many." Before you snicker, remember that the concept of number has nothing to do with the size of a number vocabulary. Whether or not people know words for big numbers (like "four" or "quintillion"), they can know that if two sets are the same, and you add 1 to one of them, that set is now larger. That is true whether the sets have four items or a quintillion items. People know that they can compare the size of two sets by pairing off their members and checking for leftovers; even mathematicians are forced to that technique when they make strange claims about the relative sizes of infinite sets. Cultures without words for big numbers often use tricks like holding up fingers, pointing to parts of the body in sequence, or grabbing or lining up objects in twos and threes.

Children as young as two enjoy counting, lining up sets, and other activities guided by a sense of number. Preschoolers count small sets, even when they have to mix kinds of objects, or have to mix objects, actions, and sounds. Before they really get the hang of counting and measuring, they appreciate much of its logic. For example, they will try to distribute a hot dog equitably by cutting it up and giving everyone two pieces (though the pieces may be of different sizes), and they yell at a counting puppet who misses an item or counts it twice, though their own counting is riddled with the same kinds of errors.
Formal mathematics is an extension of our mathematical intuitions. Arithmetic obviously grew out of our sense of number, and geometry out of our sense of shape and space. The eminent mathematician Saunders Mac Lane speculated that basic human activities were the inspiration for every branch of mathematics:
Counting -» arithmetic and number theory
Measuring —> real numbers, calculus, analysis
Shaping —> geometry, topology
Forming (as in architecture) —> symmetry, group theory
Estimating —> probability, measure theory, statistics
Moving —> mechanics, calculus, dynamics
Calculating —> algebra, numerical analysis
Proving —> logic
Puzzling —» combinatorics, number theory
Grouping —> set theory, combinatorics

Mac Lane suggests that "mathematics starts from a variety of human activities, disentangles from them a number of notions which are generic and not arbitrary, then formalizes these notions and their manifold interrelations." The power of mathematics is that the formal rule systems can then "codify deeper and non-obvious properties of the various originating human activities." Everyone—even a blind toddler—instinctively knows that the path from A straight ahead to B and then right to C is longer than the shortcut from A to C. Everyone also visualizes how a line can define the edge of a square and how shapes can be abutted to form bigger shapes. But it takes a mathematician to show that the square on the hypotenuse is equal to the sum of the squares on the other two sides, so one can calculate the savings of the shortcut without traversing it.

Consider this request: Visualize a lemon and a banana next to each other, but don't imagine the lemon either to the right or to the left, just next to the banana. You will protest that the request is impossible; if the lemon and banana are next to each other in an image, one or the other has to be on the left. The contrast between a proposition and an array is stark. Propositions can represent cats without grins, grins without cats, or any other disembodied abstraction: squares of no particular size, symmetry with no
particular shape, attachment with no particular place, and so on. That is the beauty of a proposition: it is an austere statement of some abstract fact, uncluttered with irrelevant details. Spatial arrays, because they consist only of filled and unfilled patches, commit one to a concrete arrangement of matter in space. And so do mental images: forming an image of "symmetry," without imagining a something or other that is symmetrical, can't be done.

The concreteness of mental images allows them to be co-opted as a handy analogue computer. Amy is richer than Abigail; Alicia is not as rich as Abigail; who's the richest? Many people solve these syllogisms by lining up the characters in a mental image from least rich to richest. Why should this work? The medium underlying imagery comes with cells dedicated to each location, fixed in a two-dimensional arrangement. That supplies many truths of geometry for free. For example, left-to-right arrangement in space is transitive: if A is to the left of B, and B is to the left of C, then A is to the left of C. Any lookup mechanism that finds the locations of shapes in the array will automatically respect transitivity; the architecture of the medium leaves it no choice.

Suppose the reasoning centers of the brain can get their hands on the mechanisms that plop shapes into the array and that read their locations out of it. Those reasoning demons can exploit the geometry of the array as a surrogate for keeping certain logical constraints in mind. Wealth, like location on a line, is transitive: if A is richer than B, and B is richer than C, then A is richer than C. By using location in an image to symbolize wealth, the thinker takes advantage of the transitivity of location built into the array, and does not have to enter it into a chain of deductive steps. The problem becomes a matter of plop down and look up. It is a fine example of how the form of a mental representation determines what is easy or hard to think.
— Steven Pinker
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You seem to be conflating "performing" with "thinking". Does Justin "think", or "perform"? Is there a difference between "performing" mathematical calculations as opposed to "thinking" of mathematical calculations?

Well, if you must make a distinction between performance and thinking it must be mean that the former doesn't involve the latter and also the converse. That makes sense for the reason that normal physical activity tends to occur at a level of neural activity that doesn't register in consciousness which my guess is what you mean by thinking. However, as far as my argument is concerned, "thinking" means everything that occurs in the brain, whether our consciousness is aware of it or not.
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