# Big math/science results within easy reach

Regarding interesting mathematical results involving rationality (in a different, economics sense) you may wish to have a look at Arrow's Theorem:

A non-technical summary: if a decision must be made by more than two people or even if a single decision maker uses more than two criteria, then there is no way that the outcome be rational (meaning that some of the common-sense properties imposed on the preferences will be violated).

Personally, I have always found it really disturbing.

Regarding interesting mathematical results involving rationality (in a different, economics sense) you may wish to have a look at Arrow's Theorem:

A non-technical summary: if a decision must be made by more than two people or even if a single decision maker uses more than two criteria, then there is no way that the outcome be rational (meaning that some of the common-sense properties imposed on the preferences will be violated).

Personally, I have always found it really disturbing.
I've heard Arrow's Theorem mentioned from afar, never in any text I've studied.
The text you posted is unfamiliar territory for me. Hoping 3blue1brown or someone of the sort would do a good talk.

Is it generally accepted without reserve?
I ask because I've been in discussions where Gödel’s incompleteness or Turing's incomputability were deemed to apply to *machines*, yet somehow not to to humans, via some magic with the Heisenberg principle, quantum tunneling or similar magic, the fancier the better.

Regarding interesting mathematical results involving rationality (in a different, economics sense) you may wish to have a look at Arrow's Theorem:

A non-technical summary: if a decision must be made by more than two people or even if a single decision maker uses more than two criteria, then there is no way that the outcome be rational (meaning that some of the common-sense properties imposed on the preferences will be violated).

Personally, I have always found it really disturbing.
There are mathematical and scientific methods to ensure logical and consistent multi-criteria decision making. Philosophy on the other hand, is neither math, nor science.

There are mathematical and scientific methods to ensure logical and consistent multi-criteria decision making. Philosophy on the other hand, is neither math, nor science.
Just because the link was from the Encyclopedia of Philosophy doesn't mean it's not math nor science.
Arrow won the Nobel Prize in Economics, for this result among others.
Philosophers sometimes discuss math results with much interest, without tainting those results.

I think this is a better summary of Arrow's theorem.

According to Arrow's impossibility theorem, in all cases where preferences are ranked, it is impossible to formulate a social ordering without violating one of the following conditions:
• Nondictatorship: The wishes of multiple voters should be taken into consideration.
• Pareto Efficiency: Unanimous individual preferences must be respected: If every voter prefers candidate A over candidate B, candidate A should win.
• Independence of Irrelevant Alternatives: If a choice is removed, then the others' order should not change: If candidate A ranks ahead of candidate B, candidate A should still be ahead of candidate B, even if a third candidate, candidate C, is removed from participation.
• Unrestricted Domain: Voting must account for all individual preferences.
• Social Ordering: Each individual should be able to order the choices in any way and indicate ties.

So it doesn't say rational choice is impossible, only that one must violate one of the above conditions.

I'd say that Godel's theorem is pretty approachable. Pretty much the Cantor diagonal proof on steroids.

(above is a pointer to Amazon : "on formally undecidable propositions of the principia mathematica"

A bit more work is required to digest the Pea and the Sun paradox (Banach-Tarski Paradox), but once there your mind will be properly blown as the realization of how strange infinite sets can be hits home.

have fun!

jack

I'm a simple hairy-knuckled country-boy engineer
Hey, @DonH56, you made me check my knuckles!
Mine are also hairy [possibly balding].. where do I sign up!

I'd say that Godel's theorem is pretty approachable. Pretty much the Cantor diagonal proof on steroids.
View attachment 278398

(above is a pointer to Amazon : "on formally undecidable propositions of the principia mathematica"

A bit more work is required to digest the Pea and the Sun paradox (Banach-Tarski Paradox), but once there your mind will be properly blown as the realization of how strange infinite sets can be hits home.

have fun!

jack
Well, presenting a derivation on paper based on imaginary concepts (scatering of points) as something exactly applicable to real world solid objects (peas and stars) is grossly misleading.

Well, presenting a derivation on paper based on imaginary concepts (scatering of points) as something exactly applicable to real world solid objects (peas and stars) is grossly misleading.
good point. solid objects ain't solid and confusing physics and mathematics can lead to all kinds of category errors.
maybe take the pea and the Sun as a metaphor for a very small diameter solid sphere and a very large diameter solid sphere.

good point. solid objects ain't solid and confusing physics and mathematics can lead to all kinds of category errors.
maybe take the pea and the Sun as a metaphor for a very small diameter solid sphere and a very large diameter solid sphere.
Still not applicable to physical objects by any stretch.

good point. solid objects ain't solid and confusing physics and mathematics can lead to all kinds of category errors.
maybe take the pea and the Sun as a metaphor for a very small diameter solid sphere and a very large diameter solid sphere.
Does that statement bring us to the discussion of whether a [good or bad] point in every direction is really a point?
Mathematically and/or philosophically, I've never fully wrapped my head around that riddle... pro or con.

Most of the times the magic that blow one's head off is not in the complexity of the principle/theorem, but the way it's applied to much complicated problems. Take the pigeon hole principle for instance: It states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. Sound simple enough, right? It's so simple in fact that it makes you wonder why on earth would anyone bother with generalizing such an obvious fact as a principle...

Then let me ask you this, do you know the probability of at least two people having the exact number of hair on their heads at a given town? Typical human head has an average of around 150,000 hairs, it is reasonable to assume that no one has more than 500,000 hairs on their head. Now applying the pigeon hole principle:
n = town population (pigeons)
m = unique number of hairs on one's head (holes) = 500,000
If the town's population is over 500,000 it's certain that there's at least one other person with the same number of hairs on their head as another.

If you walk into a room with more than 366 others, you can be absolutely certain (enough to bet your life on) that there's at least 2 people with the same birthday.

I also just thought of when my father boggled my mind with this one:

1=.9bar

Edit for clarity.
.9bar = 0.999999999... I couldn't find a way to actually have a superscript, and ellipses I thought might be less clear, so, it is meant to represent 9s going on forever.

Nothing to do with pressure. Sorry 'bout that.
End edit.

The proof was simple enough for my young mind to grasp, but profound enough to leave quite the impression.
A bit of Zeno's paradox about the time taken to for the arrow to go 1/2 the distance to its target, and in the next interval to go 1/2 of that and so forth such that it takes infinite time to reach the target.

Most of the times the magic that blow one's head off is not in the complexity of the principle/theorem, but the way it's applied to much complicated problems. Take the pigeon hole principle for instance: It states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. Sound simple enough, right? It's so simple in fact that it makes you wonder why on earth would anyone bother with generalizing such an obvious fact as a principle...

Then let me ask you this, do you know the probability of at least two people having the exact number of hair on their heads at a given town? Typical human head has an average of around 150,000 hairs, it is reasonable to assume that no one has more than 500,000 hairs on their head. Now applying the pigeon hole principle:
n = town population (pigeons)
m = unique number of hairs on one's head (holes) = 500,000
If the town's population is over 500,000 it's certain that there's at least one other person with the same number of hairs on their head as another.

If you walk into a room with more than 366 others, you can be absolutely certain (enough to bet your life on) that there's at least 2 people with the same birthday.
Remember a teacher in my high school going on about odds of 2 people having the same birthday in numbers less than 366. We had an unusual group. He kept adding his classes to the number searching for same birthdays. You have a 50% chance of duplicate birthdays at 23 people. 99% chance at 57. We made it to 318 if I recall, which was the entire senior class with no duplicate birthdays. The odds are something in the trillions and trillions. OTOH, I've had 4 girlfriends whose father had my birthday. I've not had all that many girlfriends so quite unusual.

Also played in a friendly poker tournament once with I think 15 people. In the span of two hours there were two Royal Flush hands. Even simple statistics are weird sometimes.

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Remember a teacher in my high school going on about odds of 2 people having the same birthday in numbers less than 366. We had an unusual group. He kept adding his classes to the number searching for same birthdays. We made it to 318 if I recall, which was the entire senior class with no duplicate birthdays. OTOH, I've had 4 girlfriends whose father had my birthday. I've not had all that many girlfriends so quite unusual.

Need just 23 people for a 50% chance (from Wikipedia):

Most of the times the magic that blow one's head off is not in the complexity of the principle/theorem, but the way it's applied to much complicated problems. Take the pigeon hole principle for instance: It states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. Sound simple enough, right? It's so simple in fact that it makes you wonder why on earth would anyone bother with generalizing such an obvious fact as a principle...

Then let me ask you this, do you know the probability of at least two people having the exact number of hair on their heads at a given town? Typical human head has an average of around 150,000 hairs, it is reasonable to assume that no one has more than 500,000 hairs on their head. Now applying the pigeon hole principle:
n = town population (pigeons)
m = unique number of hairs on one's head (holes) = 500,000
If the town's population is over 500,000 it's certain that there's at least one other person with the same number of hairs on their head as another.

If you walk into a room with more than 366 others, you can be absolutely certain (enough to bet your life on) that there's at least 2 people with the same birthday.
Another spin: how many people need to gather in order for the probability of two people having the same birthday is 50%?
How many to reach a p = 0.99?
I'll provide the answers later.

Remember a teacher in my high school going on about odds of 2 people having the same birthday in numbers less than 366. We had an unusual group. He kept adding his classes to the number searching for same birthdays. We made it to 318 if I recall, which was the entire senior class with no duplicate birthdays. OTOH, I've had 4 girlfriends whose father had my birthday. I've not had all that many girlfriends so quite unusual.

Also played in a friendly poker tournament once with I think 15 people. In the span of two hours there were two Royal Flush hands. Even simple statistics are weird sometimes.
See below. Your experience was most unusual.

Sorry, looked up the info while you guys were posting. Edited my post to add the info before seeing the other posts. So we sort of posted the same stuff. That is okay however.

If you walk into a room with more than 366 others, you can be absolutely certain (enough to bet your life on) that there's at least 2 people with the same birthday.
Are there any rounding errors in your numbers or relying too much on averages?
Perhaps a minor correction for leap-seconds and/or one for the fact that more babies are born during certain seasons than others?
That would be beyond my 6-sigma tolerance!

Are there any rounding errors in your numbers or relying too much on averages?
Perhaps a minor correction for leap-seconds and/or one for the fact that more babies are born during certain seasons than others?
That would be beyond my 6-sigma tolerance!
I thought you'd point out leap year. Someone born on February 29th, and you could have 366 and no one with the same birthday.

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