Ah, but it's worse than this. The truly ambitious ladder climber creates not just unnecessarily complicated abstractions, but organizations. Processes for people to follow. Infrastructure for people to maintain. Committees to vet changes. Standing meetings.
Extraordinary claims require extraordinary evidence. Can you cite any claims by mathematicians that there were "gaps"? It isn't even true for rational numbers that you can identify an unoccupied "gap".
Yeah, it took me a second, too. By "gaps" they mean numbers that can't be represented in a given construction. So irrational numbers are "gaps" in the rational numbers, and transcendental numbers are "gaps" in the algebraic numbers. Not the best spatial metaphor.
You’re thinking of this with the benefit of dedekind in your schooling - whether or not your calculus class told you about him.
Density - a gapless number line - was neither obvious nor easy to prove; the construction is usually elided even in most undergraduate calculus unless you take actual calculus “real analysis” courses.
The issue is this: for any given number you choose, I claim: you cannot tell me a number “touching” it. I can always find a number between your candidate and the first number. Ergo - the onus is on you to show that the number line is in fact continuous. What it looks like with the naive construction is something with an infinite number of holes.
I think you are getting away from my point, which pertains to what the article said, which is that mathematicians thought there were "gaps". What mathematician? Can I see the original quote?
The linguistic sleight-of-hand is what I challenge. What is this "gap" in which there are no numbers?
- A reader would naturally assume the word refers to a range. But if that is the meaning, then mathematicians never believed there were gaps between numbers.
- Or could "gap" refer to a single number, like sqrt(2)? If so, it obviously is not a gap without a number.
- Or does it refer to gaps between rational numbers? In other words, not all numbers are rational? Mathematicians did in fact believe this, from antiquity even ... but that remains true!
Regarding this naive construction you are referring to: did it precede set theory? What definition of "gap" would explain the article's treatment of it?
I don’t know the answers to all of your questions - but I believe you’d benefit from some mathematical history books around the formalization of the real analysis; I’m not the best person to give you that history.
A couple comments, though - first, all mathematics is linguistics and arguably it is all sleight of hand - that said the word “gaps” that you’ve rightly pointed out is vague is a journalists word standing in for a variety of concepts at different times.
The existence of the irrationals themselves were a secret in ancient greece - and hence known for thousands of years, but the structure of the irrationals has not been well understood until quite recently.
To talk precisely about these gaps, if you’re not a mathematical historian, you have to borrow terminology from the tools that were used to describe and formalize the irrationals -> if former concepts about the lines sound hand-wavy to you, it is because they WERE handwavy. And this handwaviness is about infinity as well, the two are intimately connected. In modern terms, the measure of the rationals across any subset of the (real) number line is zero - that is the meaning of the “gaps”. There is, between any two rationals, a great unending sea where if you were to choose a point completely at random, the odds of that point being another rational is zero.
EDIT: for a light but engaging read about topics like this, David Foster Wallace’s Everything and More is excellent.
I think you will agree that the bulk of your comment employs a post-set-theory nomenclature.
Regarding "if you were to choose a point completely at random, the odds of that point being another rational is zero", I ponder the question of how one might casually "choose" a value with infinite entropy.
And by the way, what the Hell is up with all these people claiming that two spaces is an obsolete typewriter-era pre-proportional-font thing? Narrow proportional spaces make two spaces after a period MORE important for visually separating sentences. Is it old fashioned to think logically?
Yes, I find it useful while editing regardless of the final rendering. Maybe it's a quirk of how I process information visually, or a holdover from learning to type on an avocado green Selectric.
Funny enough, vanity sizing strikes there too. The purported waist size of a pair of Levi's is off by almost three inches.
One might argue that the size on their label is not supposed to indicate the size of the garment waistband, but the waist size of the wearer who would find it comfortable, but even with that interpretation it doesn't work out right.
Hold on there. High fiber consumption increases the excretion of cholesterol, by reducing the reabsorption of the cholesterol in bile. The liver produces cholesterol for bile, which mixes with our food in the duodenum and aids absorption of fats. Most of this cholesterol is then re-absorbed by the small intestines. By increasing bulk, fiber reduces the amount that is re-absorbed.
Effects on but biome are real too, and apparently beneficial, and may factor in, but it isn't the only (or necessarily the primary) mechanism for reducing serum cholesterol.
1g/lb is in fact a popular target among bodybuilders.
Much research indicates 0.5 to 0.7 g/lb provides most of the benefits, with continuing but diminishing gains above 0.7 g/lb. And the benefit is not just for "body building", but also for minimizing muscle loss during weight loss and improving insulin sensitivity. Other research indicates we may benefit from higher levels as we age.
