Common sayings expressed in Mathematical notation

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I have an idea for a research project which could produce a really fun paper. I think that in this case, having a lot of participants and co-authors will make it more methodologically sound.

I'd like to express a lot of different sayings or aphorisms in math. My first thought was "A bird in the hand is worth two in the bush." I'll have to figure out the easiest way to post the notation that I need on WP; advice is most welcome.

The reason for doing this is to find the original ideas people have about economic concepts, since the best model seems to indicate that they behave like Bayesian updaters to some extent. My professor seemed a little shocked when I asked where their original prior probability came from. The following week, I came back to class with the idea that we learn those priors through what people say. Maybe the classic sayings people repeat to children create their initial prior probability, which they then update as they get more experience.

Gathering different sayings from various cultures which relate to economics and lend themselves to mathematical expression, and trying to make that as consistent with the idea of the saying as possible, seems like one pathway into my research goals of advancing behavioral economics. I'd like to hear from anyone on this, if you have any ideas for useful sayings, ways to address the conversion, and ways to post notation here. Thanks in advance!

Interesting idea.

There's
"A stitch in time saves nine".

The saying has to do with staving off things ahead of time. But I have NO idea what the literal meaning is. It was explained to me once but I forgot. If you stitch a rip in garment it will "save nine" of...something.

"A penny saved is a penny earned".

"An once of prevention is worth a pound of cure".

"The early bird gets the worm".

I'm looking at a bird in the hand as one unit of consumable goods, and the two in the bush as representing the expected value of the two in the bush, given some measure of expected value. Assuming that the saying implies letting go of the one bird in order to catch the two, we have:

Let B represent the consumption value of a bird. Then B > E(2*B) represents a bird in the hand being of greater value than the expected value of the two in the bush. If we distinguish between Bh, a bird in the hand, and Bb, a bird in the bush, we can say Bh > E(2*Bb) - Bh, we represent the idea that you have to let go of the bird in the hand in order to pursue the two in the bush. Then 2*Bh > E(2*Bb). Note that the expected value of a bird in the hand, E(Bh) equals Bh. Substitute that in, and you get E(2*Bh) > E(2*Bb). Given repeated trials, the value of a bird in the bush is lower than the value of a bird in the hand. So goes the saying, and we can examine the ways that affects people's thinking.

Then there is the age old question. Not to solve it, but just to restate the question in mathematical terms.

"How many board feet of lumber can be projected kinetically through space by a medium sized rodent?"

Or...

"How much wood can a woodchuck chuck if a woodchuck could chuck wood?"

I think it's a stitch in time saves nine (stitches too late). If we make let t = T represent the time at which something reaches the point of absolutely having to be fixed, and for some positive number a, let T - a < T represent some time prior to that at which a fix could be preemptively applied, and let f(x) be a function which describes the economic cost of making the repair at some point in time x = t, we have 9*f(T - a) = f(T).

An ounce of prevention is worth a pound of cure fits the same formula, but asserts that 16*f(T - a) = f(T)

Same concept, different ratios

I don't know why I included x, above. Just think f(t) such that 0 < t < T, where 0 represents the first opportunity to apply the fix. Then f(t) is an increasing function. We might be able to determine its shape. It doesn't seem like a linear function. Maybe a step function? Then the value of f would remain at one stitch, then abruptly step up to two stitches, or possibly suddenly go straight from one to nine stitches.

I prefer to express these in terms of probability functions to capture the idea of a smoothly increasing risk function. I suspect that will more accurately portray the way people think. As mentioned in my first post, I want to extend our understanding of how people operate as Bayesian updaters. That just means that they reassess the probability of things happening based on new information. We don't understand where that starts; I mean, you have to have something to update in the first instance of updating it. There's a there there.

"too legit to quit" = All real numbers approaching infinity


"a stopped clock is right twice a day" = a side of a rhombic dodecahedron die = time, when time MOD 12 = 0

To put it into the probability expression which I prefer, let b represent bulls***, a random draw from a known bulls***ter, B. Let a represent an assessment, a random draw from a known good source, A.
Then, we have a set of conditional probabilities, P(b|B) and P(a|A) among them. Note that P(a is true|B), the probability of getting a good prediction from a known bad source is the complement of P(b|B), bulls*** from a bulls***ter, so
P(a|B) = 1 - P(b|B).

To express a stopped clock is right twice a day, for simplicity we can limit the value of right (true) to "right about the hour," although you can make it discrete however you prefer and change the result. Interestingly, if you make time continuous, then the probability of the clock being right drops to zero. Solve the integral if you doubt it.

Anyway, back in discrete, hourly time. P(a|B)*12=P(b|B)
If we refine it down to the minute, then P(a|B)*720=P(b|B) Ouch!

Let l represent the condition of legitimacy, and q represent quitting. P(q) = f(l), where there exists some L such that for l > L, P(q) = 0. Then L represents the degree of legitimacy beyond which one is too legit to quit.

Q equals the pressures to quit.

L equals the amount legitimacy gained over time.

Both grow over time.

So if you graph it: both both lines would slope upward going to right of the origin.

L would always start at the origin at zero (since a brand new artist starting out has yet to have any legitimacy).

The Q line would start out high up in the positive veritical axis above the origin (because even starting out there are lotsa pressures to quit).

But ideally the L line rises at a steeper slope than the Q line, and would eventually cross the Q line thus representing that moment that the value of L exceeds the value of Q.

And that moment would be labeled "HM" (for "Hammer time") because that would be the moment when it becomes "To legit to quit".

“The good of the many outweight the good of the few”

G - good
M - the many
F - the few

G*m > G*f

A stopped clock can NOT be represented with a 12 sided dice. That's because most clocks have two hands, an hour, and a minute hand.

For a stopped clock to agree with a working clock both hands would haft to agree on both clocks. So the dice would need 12 times 60,or 720 sides, for it represent the stopped clock.

I've seen a watch with 24 hours on the dial and only one hand. It gives a general sense of the time at a glance. It would discretize time to the level of the observer's focus. If stopped, it would be right for some fuzzy interval once a day.