Formulating Conceptual Problem Mathematically

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CharlesW

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Hello dear friends,

Many thanks to the Admins of this great site and all of the kind people who have contributed help.

My problem concerns how to reduce a conceptual problem in evolutionary biology or perhaps, cognitive psychology, to a mathematical expression.

Imagine the following situation: A young human mother tens of thousands of years ago is digging in a field for tubers, accompanied by her young infant. She hears a distant rustle in the grass and sees a lioness creeping slowly toward her. Let's call this situation, an 'interaction.' It is an interaction between the individual human and its environment, in this case, a life and death matter. As such, it is relevant to natural selection. The human mother must act. Her first thought is fear for her own life and that of her infant. She has a certain amount of brain power, or intelligence or cognitive capacity ('c'). In general, it is the cognitive capacity of her species, but of course there is variation among the individuals comprising her species. Let us set aside this variation for the moment. Think of cognitive capacity as computing power, such as the megahertz of a computer CPU. Meanwhile, all of the above information the mother has to process in order to act within the time she has is 'i.' The time within which she must act is 't.'

So we have three variables:

c = the cognitive capacity of the mother
i = the information that the mother must process
t = the time within which the mother must process that information

It seems clear that there is an inverse relationship between c and i. So I keep feeling this relationship must be similar to the gravity equation I learned ages ago, or F - Gm1m2/r^2. But I sense this is the wrong approach. I can't figure out how to place t. What I'm struggling with is how to express the relationship among these three variables mathematically.

I hope some kind person can help!

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Note: This isn't relevant to my question but I define 'intelligence' in the present context as the sum total (enhanced by the synergy) of instinct, neural processing power, the extent and accuracy of the cognitive model the individual has acquired over the course of its life (its knowledge), and heuristic knowledge (gut feeling arising from experience).

Sorry for any spelling errors.
 
If cognitive capacity is pure processing power or rate of information processing, then I would say `c=i/t`. No idea if this makes any sense. But it doesn't look like c and i are in inverse relationship - if time is fixed, the more cognitive capacity you have, the more information you will process.
 
Thanks. I see that c is essentially the _measure_ of the rate at which the information is processed expressed as a ratio as in miles per hour, or m/h. More miles per hour would increase the value of m/h by increasing the numerator. In this case, c (capacity) would represent the power of the motor of the car. Or in an egg-eating contest, it would represent the size of the contestant's stomach. The rate would be eggs eaten per minute (e/m). In these cases, an increase in c (with i held constant) results in a reduction in the denominator t, raising the value of i/t.

I think I need to try substituting some numbers for the variables to understand the relationship between these three factors a bit better. I may be back with a reformulated question. Thanks for the helpful comment.
 
CharlesW said:
Thanks. I see that c is essentially the _measure_ of the rate at which the information is processed expressed as a ratio as in miles per hour, or m/h. More miles per hour would increase the value of m/h by increasing the numerator. In this case, c (capacity) would represent the power of the motor of the car. Or in an egg-eating contest, it would represent the size of the contestant's stomach. The rate would be eggs eaten per minute (e/m). In these cases, an increase in c (with i held constant) results in a reduction in the denominator t, raising the value of i/t.

I think I need to try substituting some numbers for the variables to understand the relationship between these three factors a bit better. I may be back with a reformulated question. Thanks for the helpful comment.
Click to expand...
If information is analogous to distance, then it may be helpful to look at other ways distance can be expressed as a function of time (speed and acceleration can be variable, etc.) In terms of info processing - I doubt the relationship is that simple. Given a problem some time is need to 'register' it, then to create some sort of model in your mind that you can use to solve it, then to actually solve it. The rates of processing these subtasks are most likely different (you mentioned these types of processing in your note).
 
I suspect that you are making a mistake trying to specify functions. You probably should do as economists do and think about highly general functions.

A simple demand function is specified as this

[MATH]p = 0 \implies d(p) = a > 0; \\ \exists \ b > 0 \text { such that } p \ge b \implies d(p) = 0; \text { and }\\ 0 < p < b \implies 0 \le d(p) < b \text { and } d’(p) \le 0.[/MATH]
The economist assumes only that quantity demandEd is modeled on a function that never increases as price increases.
 
Many thanks Jeff and Lev. Thanks for your kind comments. I'm an old guy and have recently retired with the (silly) intention of resuming a long-last dream of returning to grad school, after returning to this country after nearly 40 years. My undergrad major was economics, and be sure Jeff I will study your kind advice, but you're talking above my head, in fact. I will study your post and be back, I hope, within a month or so.

Many thanks dear friends.
 
I apologize for using what may seem to be obscure language.

My primary point is actually simple. When you are asking yourself how some process works and you have virtually no quantitative data to rely on, do not try to formulate highly specific mathematical models. Such models are highly likely to be wrong. Instead, try to formulate the most general model you can.

I used economics as an example. Look in any beginning text on economics. The text assumes that the quantity demanded falls as price rises. (There is no discussion about circumstances when quantity demanded increases as price increases.) In a more advanced course, it would be noted that prices are not negative, that if something is desirable at all, a positive amount of it will be demanded whenever it is free, and that there is some price at which none is demanded at all. These are very general attributes and can be described mathematically, letting us use mathematical tools.
 
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