Okay - I’ll admit that might have been a click-baity caption (at least in this here neighborhood) just to draw you into the most trivial of potentially weird things to ponder, but here it is…
As I was doing something productive (working on Geometry stuff for school), I found myself pondering this question:
If three lines are randomly plunked down on a plane, are they guaranteed to form a triangle?
The trivial answer to it is: no. If any of the three lines are parallel, then no triangle is formed. Or likewise they could all converge on one point, like spokes.
But what is the probability of these non-triangular occurrences happening? Well - for perfect lines, the probability of anything being perfectly parallel, much less all coinciding at a point of concurrence is precisely zero! So statistically speaking, you have a 100% chance (certainty, actually) of getting a triangle. And yet logically speaking, the non-triangular scenario singularities do exist - (and in fact there are an infinite number of them too - it’s just an infinitely smaller infinity). So it can’t be considered as ‘proven’ that a triangle must occur - because of the existence of those counter-examples. And yet, there it is: 100% certainty in all its glory, thumbing its nose defiantly at anybody who asks for proof!
So … we’ve talked about different levels of ‘wrongness’ - so many creative ways to be wrong. But here on the flip side, do we actually have different levels of valid certainty? Certainty with proof, but yet also valid certainty even without it! And no - this has nothing to do with apologetics, though I recognize how the religious mind can’t resist taking it there. Do so if you must. I’m just thinking about something interesting here - as a great way of avoiding productivity here at the moment.
I think, perhaps, the apparent paradox is one of semantics. In statistics, a continuous variate, ie. one whose values are in the continuum such as the real numbers, always has probability 0 of taking on a specific value. So, in your case it is the probability of the angle of incidence between two random lines being a specific value, 0 degrees, meaning they are parallel. The probability of this is 0. For continuous variates, it only makes sense to work with a probability density function, or pdf. This gives the probability that the value lies between a and b, for values a < b. So, we work with the probability of the angle of incidence of two random lines falling between 0 and 0.001 degrees, for example. That would be small, but non zero. It’s only in discrete variates, where the values fall in a range of discrete values, does it make sense to have a probability distrubution in which the variate takes on a specific values.
Of course, in practice we have measuring devices of only finite accuracy, so variates may be discrete in practice. But it often makes sense to deal with continous variates.
Sorry I don’t have a deep theological take on this. Does God operate in the continuum? Or, is that a human-made concept? Perhaps God made the universe in discrete chunks, of unimaginably immense granularity – far beyond human comprehension, but still discrete.
I imagine the above video talks about Georg Cantor, and the set theory of infinities.
It may help to tighten the language a bit. When you say the probability is “100%,” what that really means in mathematics is almost sure, not logically certain. In your triangle example, the “bad” cases (parallel lines, all meeting at one point) don’t disappear—they’re still there—but they form a set of measure zero. So they’re not impossible, just negligible in a probabilistic sense.
That’s why it feels like “certainty without proof,” but it’s actually something different. It’s not a second kind of logical certainty—it’s a different framework altogether. In logic, certainty means: cannot be false. In probability, “certainty” (probability = 1) means: false cases exist, but are vanishingly rare.
So the triangle result isn’t proven in the strict logical sense—it’s just overwhelmingly expected under the model you’re using. I’d say what you’ve uncovered is less “two levels of certainty” and more “two different meanings of certainty” that we tend to blur together in everyday language.
The answers to your questions depend on the area of knowledge or experience you’re talking about, don’t they.
I watched @Jammycakes’ video about infinity. There are all sorts of logically certain statements that can be made about the concept of infinity based on tested ideas that I think would be call “proofs” in the math world.
In the rest of life we operate as if we have different levels of valid certainty. Really, we are forced to for practical purposes, aren’t we?
We can hyperfocus on examples of testable/tested medical examples (vaccinations perhaps) and talk about effectiveness and dangers based on statistics. But that isn’t PROOF. It isn’t absolute CERTAINTY.
But in the day to day, we have to rely on the expectation that things work as usual, whether there is proof and certainty or not.
Good points - one can only speak meaningfully of density and range, and not of probabilities of discrete numbers in a continuum. Geralamo Cardano is a fascinating figure to me since he was the first to think of probability as a quantity between zero and one. He used it to his advantage as a gambler, and also (like so many other brilliant figures of those times) seemed to be a polymath. What he did as a physician is still illuminating of so many human attitudes even today.
In a logarithmic sense. Exponents can go to negative infinity in order to investigate the infinitely small (an infinitesimal). But strictly speaking in calculus, “negative infinity” would still be a vast magnitude (large numbers) but just in the negative direction).
And this is one of the problems with talking about probability where there is no population – you build in assumptions about what “random” means. The truth is that the probability depends on how the so called “random” lines are generated. In science we start with a population from which we can calculate probabilities or deduce the process by which “the lines” might have been generated. Consider for a example of picking your random lines from lines which people have drawn/created. In this case the probabilities of 3 lines not forming a triangle are far from zero.
Good points. That practicality (as you go on to discuss regarding medical things or vaccinations) is where the rubber meets the road! It is where we live after all.
I was envisioning taking three ‘pick up sticks’ like I used to play with as a kid, and giving them a “twirling toss” down onto a flat clear table top. That would then become the randomly placed line on that plane. But in architecture (or geometry texts), lines are anything but random as you say!
Properly speaking, they are the same infinity (both can be put into one-to-one correspondance with the set of all real numbers) but one is infinitely less dense.
That sounds plausible to me - they are both ‘c’ (Aleph 2), I suppose, and as such have the same cardinality then - as the speaker in @jammycakes video makes clear.
So that then just highlights the question for me: should there be ‘subrankings’ within one domain of countability? The fact that even within countable infinities, you still have an ‘infinite range’ of comparative infinities (I guess like how the rationals are infinitely more dense than the mere counting numbers - and yet they both rank as equally denumerable infinities).
I hope to yet watch the follow up video for that, James. Thanks for sharing the first one.
@Mervin_Bitikofer This is excellent, for reasons you probably do not know.
If you have a sheet of graph paper and randomly choose 3 pairs of points to draw lines, you could calculate the probability of parallel lines as a function of the size of the graph paper.
The part you probably didn’t know, is this closely matches the definition of a Probability Space, which is a set of discrete events like the intersections on graph paper. Probability is defined on discrete events, and we “smooth over” the gaps when we take an integral over a continuous variable. (Technically Lebesgue integrals, but I try not to frighten myself with such things any more.)
