3 Fascinating concepts in Math & Physics that everyone can appreciate
An admiration for some unique nuances to scientific facts makes the lay person and not just the experts appreciate the beauty of science & math
The speed of light and what it means for how light experiences time
These days the discovery and pictures of objects close to the dawn of the universe is all the rage with the JWST. We can see stars & galaxies from as far back as 10 billion or more years ago. This means we are seeing their light reaching us now after traveling for 10 billion years at — Oh well, “The speed of Light”, the cosmic speed limit. Thus far nothing too profound for the knowledgeable amongst us.
But what is seldom mentioned explicitly or pondered over is this — How does light itself experience this journey of 10 billion years?
Well, the short answer is it doesn’t!
As Einstein taught us with Special Relativity, at the speed of light, time comes to a halt. Light only ever travels at the speed of light and so it does not experience any passing of time. So, no matter how far light travels, from its point of view it travels instantaneously even though the speed at which it travels, though very fast, is still finite. Let that sink in for a bit.
Infinity is not a number but a concept & all infinities are NOT the same size, and some infinites you would NOT expect to be the same size ARE the same size!
We are all taught that there are sets and subsets, and that subsets have fewer elements in them than the sets they are a part of. This, we can all agree on in normal conditions.
However, strange at it seems this does not apply to infinite series. That is to say — a set of all positive integers is NO bigger than a set of just the even integers! — intrigued?
So, consider the positive integers — 1,2,3,4,5 …
Now consider only the even integers — 2,4,6,8,10…
If both are unending infinite series, we can form a 1:1 correspondence of the form X -> 2X thus proving that these two infinites are of the same size. This is called the cardinality rule.
Even more mind boggling is the fact that the set of all number between 0 and 10 is no bigger than a set of all numbers between 0 and 1. Infinities can be perplexing and defy common sense, right?
Just when you are beginning to think that all infinities are the same, here’s a curve ball for you — which is bigger 1) The set of all numbers between o and 1 or 2) The set of all integers?
Some infinities are countable, and others are not. Mathematician Gregor Cantor proved using the diagonal table method that the set of numbers between 0 and 1 are not countable I.e., they cannot be placed in a 1:1 correspondence with the set of integers hence proving that the set of number between 0 and 1 is indeed larger than the set of all integers
Bayesian Probability makes diseases test diagnoses rather more palatable
Here is a critical life moment where intuition fails you utterly, creating unnecessary angst, worry & suffering:
You undertake a test for a rare disease that is highly accurate, and you get a positive result. You are devastated. It’s a highly accurate test so you must be highly likely to have this rare disease, right?
WRONG!
This is where knowing Bayesian probability stands you in good stead. Thomas Bayes proved that a priori probability (or simply priors) plays a critical role that is often ignored by the uninitiated.
Consider an example — There is a rare disease that afflicts 1% of the population. There is a test for this disease that is 90% accurate. You take the test, and it comes out positive. Extremely bad, right? You are almost guaranteed to have this disease, right?
Not so fast!
Even with this seemingly highly accurate test you are only about 8% likely to have this disease when you test positive. What a sigh of relief!
How so? — Well, the rarity of this disease in the general population is a critical and often overlooked statistic that turns intuitive and conventional wisdom on its head.
Bayes showed that:
The probability of having the disease (D) if you tested positive (+) IS NOT the same as the probability of testing positive (+) if you have the disease (D)
Bayes’ Theorem states:
P(D/+) = [P(+/D) x P(D)]/P (+)
P(D/+) = Probability of having the disease if you test positive (what we are after; needs to be calculated)
P(+/D) = Probability of testing positive if you have the disease (what we are most often given)
P(D) = Probability of the disease in the population (its rarity or otherwise — a crucial input that is estimated or given)
P (+) = Probability of testing positive (usually needs to be calculated)
So, if a test is 90% accurate in identifying a rare disease that afflicts only 1% of the population, then if you tested positive your chances of having the disease applying the above formula is:
P(+/D) =0.90
P(D) = 0.01
P (+) = P(+/D) + P (+/Not D) = .90 x .01 + .10 x .99 = 0.009 + 0.099 = 0.108
The answer is — (0.90x0.01)/.108 = 0.833 or 8.3%
So, a positive result from a 90% accurate test is not a 90% chance of you having that rare disease. Many have worried unnecessarily about the inevitability of so-called accurate test to prove certain outcomes!
Conclusion
Not everyone is a scientist or an expert with numbers. However, some acquaintance or familiarity with these concepts will make life richer, make us ponder the beauty of nature and might even make us sleep better at night knowing a positive test result for a very rare disease is not a death sentence.
