Re:0.999...= 1
by happythomist on Tuesday April 28, 2020 @01:31PM
(#23009474)
> It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas)...
Aquinas specifically objected to the notion of an essentially ordered infinite causal series. He had no objection to an accidentally ordered infinite causal series or other kinds of infinite series.
This distinction is extremely important for the purposes of understanding his proofs of God's existence, and people often unfairly reject his arguments because they conflate the two.
More reading here: http://edwardfeser.blogspot.com/2010/08/edwards-on-infinite-...
Couldn't you formulate a problem for extreme decimal-place accuracy based on the multiplication of errors in a physical process that's repeated in ways that multiply small errors into bigger ones?
1. People have had it drilled into their heads that humans can't comprehend infinity. It was taken for granted by philosophers, that an "infinite regression" is a logical fallacy (e.g., used in a proof by Thomas Aquinas), and that tricks such as infinity and the infinitesimal were not rigorous. Mathematical infinity has been a settled matter for all practical purposes since the early 20th century AFAIK.
2. Related to the above, most people also believe that there is always a gap in any knowledge, and something hiding in that gap. Thus it's perfectly natural to believe that there's something hiding between 0.999... and 1, that we just haven't found yet. Knowing for certain that there is nothing between 0.999... and 1 is regarded as a kind of arrogance.
I think the way to approach this with children is to teach math as an abstract topic, that's not necessarily rooted in the objects of everyday life. For instance there's no physics experiment that can test the necessity of any math being carried beyond roughly the 15th decimal place. Yet we enjoy exploring it anyway.