It’s unclear to me how central this is to your thesis, but I think you need to consider whether a human is really doing anything different than a dog when the human catches a ball. I’d say that, lacking clear evidence otherwise, one must assume that the human is not doing anything special.
Governed by the rules of number? Well that’s taking a pretty strict view of reality, isn’t it? How do you suppose we would come to find that something is “governed by the rules of number”? Would we have to know math first? But that would be strange…
Why? Because the ball isn’t “governed by rules of number” or because the person doesn’t know enough symbolic math to express it on paper?
Trial and error, yes, my use of “induction” indicated this process. The point is: when do we know we’ve succeeded? Are we setting roughly logical constraints on ourselves? “Hmm, every time I touch this, I get burned. He’s like me; if he touched this, he’d get burned, too.” Is this person developing a neural response, or is he doing logic, thinking logically, even if no one has ever explained implication, identity, and various logical deduction tools to him so that he may satisfy himself of the modus ponens? Why would you think the two are necessarily distinct?
The last is, there is an isomorphism between neural functions and mathematical rules as done formally.
Trusting math is not “emotional”. Again: how did we come to suggest there are rules of number?
A pattern? A pattern of what? Of thrown balls governed by gravity? Of objects dropped from tall buildings? Of the relationship between cause and effect?
I am asking you to distinguish public certainty, i.e.-knowledge, that is, what we can demonstrate to others’ satisfaction, from a feeling of certainty which can by definition only assure the person who feels it.
Yes! Exactly.
Yes, that is one kind of sharing. But sharing knowledge means exhibiting public behavior in such a way that someone else will come to know what you know. This is a subset of all sharing (ie public) behavior.
No one is compelled to be certain of something. They may reject all conventions. Intuitionists, for example, aren’t certain of any result derived from the law of the excluded middle. My personal satisfaction with that rule does not compell them in any way.
Heh! They are, aren’t they? Because the proposition about pi is a mathematical one. It must be certain, if anything is to be certain, because math is how we express certainty.
Then where did the model come from?
Sure, and my typing English has nothing to do with my neurological activities, either. :rolleyes:
All reality, except my neurons? Oh, wait, I see, it is no coincidence that my neurons can “calculate” correctly. Interesting that you seem to be agreeing with me here, though I would not use those words.
How we test for understanding of number is different than how we test for understanding of heavier. Yet the inductive or deductive processed used is logical ie mathematical in nature. When we begin to speak that language, the language which tells us such things are sure, we are able to express more and investigate more.
And I’m saying: I think you need to consider whether a human is “really” doing anything different when tracking the path of a ball than describing its path on paper with formulae. Why does that work? Why are we sure of it? What are the limits, and why are we sure of those? How do we express this certainty so that others may share it?
Clear evidence like the whole of mathematical investigation, ballistic missles, or trebuchets? Or did you mean something else?
I think I see a way to make myself at least marginally more clear. For the example that keeps coming up, a thrown ball. I get the feeling that my objectors are picturing me suggesting that the person is actually “somehow” doing calculus or algebra in his head, when that implies a certain psysiological activity. I do not mean to imply that at all. In fact the last I heard, animals catch things like balls by following, apparently, a very simple routine that seeks to keep certain distances the same (like where the ball is in one’s field of vision). That’s all very interesting scientifically, but that is really not where I’m going. I’m saying, as a person, I know where the ball will land. One way to demonstrate that certainty is to catch a ball thrown at various angles and speeds to someone’s satisfaction–and what satisfies this other person is logical in nature. Another is to plot the course of the ball as with analytic geometry, work backwards from accelleration to a position plot with integral calculus, record a bunch of data and find a function that best fits it, etc. In either case, the person is satisfying mathematical interests, one on a broadly logical matter, another on a narrowly symbolic matter. Thermostats and dogs, as a rule, do not try to convince me of anything, as far as I can tell.
I’ve admittedly only skimmed this thread, and I don’t really intend to enter the debate, but I do have a couple of things that I thought could be added. In all honesty, I don’t know that they’re particularly relevant to the debate (in the sense that anyone’s arguments will hinge one way or the other based on this), but I definitely think they are of general interest.
The first is the article Do Dogs Know Calculus?, by Timothy Pennings. To summarize, I’m sure anyone who has taken calculus at one time has seen the following problem: You’re on one side of a river, and you need to get to a point on the other side (not neccesarily the point directly opposite). You can run on land at some given speed, and you can swim at some given (slower) speed. If you run along the bank of the river, then jump in and swim directly to the intended point, at what point should you jump in the river, in order to minimize the time taken? Well, this guy Pennings performed an (informal) experiment of throwing a ball into Lake Michigan, to have his dog fetch it. The point(s) at which the dog jumped in the lake were remarkably close to the “optimal” point!
The other is something I read some time ago (sorry, no cite, this was 10 or 15 years ago, and I don’t remember what the source was). The article stated that ball players (and presumably people and dogs in general), catch fly balls by chasing it in such a manner so that the ball (in flight) appears (to them) to travel in a straight line.
erl: * Are we setting roughly logical constraints on ourselves? “Hmm, every time I touch this, I get burned. He’s like me; if he touched this, he’d get burned, too.” Is this person developing a neural response, or is he doing logic, thinking logically […]*
Well, consciously verbalizing your reasoning as informally logical propositions isn’t quite the same thing as unconsciously following neural signals that help you catch a ball.
Sure, you could make the same sort of logical chain for ball-catching, along the lines of “Hmm, when it goes up higher it takes longer to come down, when the upward curve is steeper it doesn’t go as far as when the curve is shallower”, etc. etc. That still wouldn’t be calculus, as you note, but it would indeed be conscious logical reasoning. But you don’t need even that much conscious reasoning to successfully catch a ball, and neither does a dog.
You’re saying that ability to correlate symbolic manipulation to neurological certainty is indicative of choice. I’m saying it’s indicative of ability.
Depends on the culture. Before agriculture, I would say, easily. Today, around 5-7,000. How many times have you heard eudaemonia or catapedaphobia?
Yes, this is very good! This is an observation I think is critical. But now I want to suggest: yet I can become conscious of my ability to catch a ball. How do I satisfy myself and others of this ability, consciously? “Trial and error” is an inductive test, failing to “match” mathematical induction because it is impossible for us to enumerate the entire set of thrown balls (something else we intuitively knew but came to formalize in various ways). “Doing math” is meant to imply “of or being certain in a public way”, that is, a way which could transfer certainty. Symbolic math, done on paper, is the tip of the iceburg, and we got to do symbolic math by investigating our own certainties.
You are absolutely right. Mathematics, as a field of interest, has taken on a life of its own, much as has linguistics, or history, or natural investigations (ie science). Yet we cannot break math away from its source, what caused us to create the symbology, why we find it so important (and so sure), without losing the essence of math itself as a language of certainty.
I’m saying that the recognition that I have that ability, and how I demonstrate to others that I have that ability, is mathematical in nature.
I do not see any reason to accept this at face value, largely because the written word, which did serve to increase vocabulary, came from the spoken word, which was logically and historically prior.
Why would that be a test of how many words one knows?
And anyway. II Gyan II, I don’t see whether it matters that hunter gatherers have a much smaller vocabulary than, say, a medieval knight or a twentieth century high school student. The point is that our investigations into life, and our ways of living, gave, imposed, or otherwise was correlated with our increase our vocabulary. Our investigation into farming, consciously directed or intrinsic ability based on contextual need, led us to words about agriculture. Our investigation into certainties led us to mathematics; i.e.-mathematics is the language of certainty. But language is also an activity, and so doing math is the the use or transfer of such certainty.
Because the written word codifies and preserves in memory the spoken words. If I give you 3000 words to orally remember, odds are, you will remember barely a fraction of them. You’ll remember those which are most relevant in your daily activity. When you can sustain a precise recollection via writing, then you have the leisure and ability to pick up where you left off. Which allows greater and refined expansion, not feasible before.
I’m not debating your particular ability to remember words, given that you are literate. Your original statement was “it is an abuse of the English experience to suggest that people don’t know English because they are illiterate”. Like I asked earlier, what it does mean to know something? If someone illiterate wanted to express the concept signified by eudaemonia, they would have to express it as best they could with existing vocabulary and hope that the receiver grasps the essence. Do they know English, in this case?
I think I can explain what erislover is trying to get at. Math isn’t a magic sky pixie. Math is a mode of expression of facts of the universe but it isn’t a required fact of the universe anymore than English is. There is no secret reality of computation where the math is being done to create what we see.
What this means is that math is a language. My minor quibble is that it isn’t a language like English, and I mistakingly used spoken language when I suppose social language would be closer to what I meant and not as loaded with regards to the debate. Math is a language that describes principles we hold to be required principles that we are certain of. Social languages are those that describe our social reality. I think that it is entirely possible that eventually math could develop to the point where it could be a social language as well. People could think in an endless stream of math about everything happening during their day. Math is not like that yet and I don’t think that math is wired up in the brain in the same way that social languages are. I think it is wired with similar tricks and exploitations of existing structures that language uses but the structures in question differ.
I somewhat agree with this. But I disagree that math is not necessary. The mind is not a tabula rasa, but it doesn’t come inbuilt with the neurological correlate of the solutions of a quadratic equation. The symbological expression of math can induce new understanding. So, someone who isn’t well versed with symbolic manipulation can possess deficiencies in the ultimate fact and thinking process.
Right out like that, perhaps. It depends on the context: the meaning of a word is its use, not that it is presented in a list, whether spoken or written.
Fine. Can we get to the point where this relates to my OP?
Like I answered earlier: an ability to demonstrate one’s knowledge. If the test of “knowing English” to you is that someone can express the concept of eudaemonism, first you must explain just what that concept is; that is, you must teach it to them. Then they will attempt to show that they know the concept by pointing out instances, using it in a long sentence, deriving other things from the thought, etc, to the point where its use is best served as a word rather than a phrase or sentence, at which point etc. etc… When do they “know” about eudaemonism? Doesn’t it depend on who is asking, and when and why they’re asking?
I would say so. The test of “knowing English” is not, for me, a vocabulary matter any more than the test of “thinking logically” is being able to construct any particular symbolic tautology. A greater vocabulary lends itself to a greater application of language. There is no special cut-off point where we say, “There! Now he knows.” He meets the questioner’s explicit or implicit (unconscious) conditions. The socialization of these conditions, and their application and character, results in a language, a way of doing, a way of living, thinking, speaking, and lastly: writing.
The Tim, very well said.
No, I don’t think so either, but that is a matter for natural science to investigate. Whether we are wired for it or not is outside the scope of this thread, at least as I intend it. I won’t discourage such thought, though. 
I can’t ever “know” if they know about eudaemonism. Just make a guess and be internally satisfied.
Well then no one “knows” anything and the word “know” is useless. I prefer to keep my words useful, if you don’t mind. 
They know when they can demonstrate it to me in a way that either would let me know, or would satisfy me that they know. This is the point I am attempting to drive home, not some strictly analytical binary-valued state of “knowing”, but a social phenomenon involving the investigation, conscious or otherwise, of our knowledge and certainty.
If that does not encapsulate the word “know” for you, perhaps you could do me the favor of telling me what word or concept it does match for you so that we may continue.
True for the first part and kinda true for the second part. At the very least, you have to know your memory and brain is functioning according to your assumption. But if it isn’t, how would you “know”? :D. The solution is make an assumption and hold that as truth, i.e. axioms.
This is what I said earlier: Just make a guess and be internally satisfied.
Well I think this is an abuse of the word. Honestly. I distinguish the feeling of certainty, which satisfies me and me alone, from the phenomenon of certainty, which satisfies me and those in my environment, to some degree.
But I am not guessing someone knows. I say that they know; I attribute the ability to demonstrate mastery of the concept because I have just seen them demonstrate the mastery of a concept in such a way that those who were interested in such satisfaction would also be satisfied. I’m not guessing anything.
Unless you get in their heads and become them, how do you know?
To quote my abstract algebra teacher, “This isn’t math, this is religion.” 
…seriously, now, I think math is a simplified way of looking at the world. From the most basic counting numbers (1 = “one of something”) to calculus (the ball problem) to the really abstract linear analysis business (the rules of math itself). The simpler the thing we’re analyzing, the simpler the math, and the more complex, the more equations. And besides, there’s also statistics, probability, and fuzzy logic, all of which are branches of math that deal with uncertainty.
A football player can throw a football without knowing that the spin stabilizes the fall flight; a physist can describe all the relevant equations, but not be able to throw worth beans. However, a pilot better know at least some aerodynamics if he wants to fly a plane, or he wouldn’t know how to deal with all the variables involved. (Of course, nowadays computers are doing most of the number-crunching for us, but I digress.)