Science and Inquiry discussion
This topic is about
The Outer Limits of Reason
Book Club 2023
>
August 2023 - Outer Limits of Reason
date
newest »
newest »
message 1:
by
Betsy, co-mod
(new)
Jun 28, 2023 04:44PM
Mod
reply
|
flag
I started this book by reading Chapters 1 and 2, then I jumped to Chapter 7 Scientific Limitations. My take, based on my reading is that this book is mistitled. I think it should be titled "The Outer Limits of Reason: What LOGIC (bold font), Mathematics (Normal font), and Science (very small font) Cannot Tell Us. What do you think? Did I miss a significant discussion of science topics in my abbreviated reading?
Trudy: I am currently on Chapter four and have not gotten much science yet. You are totally correct - Heavy logic and could have added Philosophy into the title. I am enjoying the book since I was on the search for a good book on logic and developing some logic skills. Let me see what I learn.....
I agree with Trudy's assessment. It even gets way deeper into logic towards the final chapters to talk about Gödel's Incompleteness Theorems and concepts like Parikh sentences.
I finally started it, and am really liking it. Started to get a headache around the part he’s going into Km is false for all I > m, got through it but I can see this is going to be slow going. These are some brain muscles I haven’t used for a long long time. Fortunately I like the material.
First I must say that I REALLY like this book. Now…I’m already having a problem with it, and it’s a problem I’ve experienced before.
In Chapter 3 there is an example:
“Look at the following infinite sun:
1/2 +1/4+1/8+1/16+1/32 …
The uninitiated would say that the ellipsis is going on forever, and so the sun total will be infinite. However, the sum total is the nice finite number 1.”
I would say the limit is 1 but it can never quite get there. Unlike the quantity of movements it takes to get from point A to point B, you could go on all your life with this and never ever reach 1. There is no limit to the number of digit you can come up with.
Certainly most of us after a few years of this would throw up our hands and say for all intents and purposes let’s call it 1. Cheat and move our foot a fraction more than Zenos rule would dictate. But mathematically it will never really be one.
Prove me wrong!
Mod
The simplest proof that I've found that 1/2+1/4+1/8+... = 1 can be found on Wikipedia:
https://en.wikipedia.org/wiki/1/2_%2B...
A simple proof that 1+1/2+1/4+1/8 + ... = 2 can be found in the Khan Academy lesson here:
https://www.khanacademy.org/math/ap-c...
Simply substitute a=1 and r=1/2.
It's a bit ironic that I found in one proof a reaction similar to yours:
Calculus Student: "Lim(n→∞)SUM[s_n] = 0 means the s_n are getting closer and closer to zero but never gets there."
Instructor: "ARGHHHHH!"
You rigorously prove the infinite sum = 1 by using the Archimedean principle: define
a_n = (1/2)^2
then the Lim(n->inf) |a_n - 1| = 0
You can find the proof using this principle in many places, for example, here:
https://homepages.math.uic.edu/~saund...
Yikes! Above my paygrade, this book. But I am giving it a go….
aPriL does feral sometimes wrote: "Yikes! Above my paygrade, this book. But I am giving it a go…."It definitely has been blowing the cobwebs off my brain....LOL; very interesting content. I am currently about halfway through chapter 7 and am particularly enjoying this section-quantum mechanics and uncertainty principle. Onward and upward.
Nancy wrote: "First I must say that I REALLY like this book. Now…I’m already having a problem with it, and it’s a problem I’ve experienced before.
In Chapter 3 there is an example:
“Look at the following infini..."
tHANK you David. I am still stuck on the "ARGHHHH!" phase. I'm grinding through this but still not convinced. I can see where for all intents and purposes (or for all intensive purposes as this is giving me a headache) we will call it "1" but to say it reaches 1 implies that there are no more 1/2^n 's we can add and I can't see where we are ever going to run out of n's which is what must happen if we hit to "1" mark. I still on the limits of sequences part. As far as Zeno's arrow goes, obviously it hits its mark. Does that mean that planck's length is truly the smallest length possible before a little vector splits into just 2 points? Has anybody ever proven this?
