Occam's Razor vs. Principa Mathematica

What is the basis for reason? And mathematics?

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roydop
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Occam's Razor vs. Principa Mathematica

Post by roydop »

The principle mathematical "proof" that 1+1=2 is 360 pages. Occam's Razor cuts that to: "One plus one equals two because we all believe that it does."

Anything that a 5 year old can understand should not require 360 pages concluding in

http://www.ivica-crnkovic.net/1+1=2.htm

to express this truth.

This looks like a programming language. A program designed as instruction for human consciousness.
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Sculptor
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Re: Occam's Razor vs. Principa Mathematica

Post by Sculptor »

roydop wrote: Wed Jun 09, 2021 2:45 pm The principle mathematical "proof" that 1+1=2 is 360 pages. Occam's Razor cuts that to: "One plus one equals two because we all believe that it does."

Anything that a 5 year old can understand should not require 360 pages concluding in

http://www.ivica-crnkovic.net/1+1=2.htm

to express this truth.

This looks like a programming language. A program designed as instruction for human consciousness.
My feelin gis that 1+1=2 because 2-1=1. That by convention the abstract values we attribute to this symbols is agreed upon by reference to other symbols in the system.
If Russell really feels he needs to go on a circular journey to prove what he already knew already then it is no wonder that in later life he dropped the idea that logic and maths could and should be integrated.
alan1000
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Re: Occam's Razor vs. Principa Mathematica

Post by alan1000 »

There is so much to unpack here, it is difficult to know where to begin.

It is important to remember that before Principia Mathematica, the whole edifice of 19th Century mathematics was built upon sand. This is why PM expends so many pages on clarifying and justifying basic definitions and axioms. That work having been done, it is now possible to economise, and we can now prove that 1+1=2 as follows:

1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2

- where S represents the successor function S(), evaluating to the number which 1 greater than the number specified in the argument, and the first 3 natural numbers are defined as 0, 1, and 2.

I recommend the opening chapters of Russell's "Introduction to Mathematical Philosophy", which is essentially PM expressed in language the intelligent lay person can understand, and where these ideas are set out very clearly.
promethean75
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Re: Occam's Razor vs. Principa Mathematica

Post by promethean75 »

Whenever my head gets too big from beating up philosophers at some forum, I find the nearest math thread so I can purge myself of excess pride and atone for being such a badass.

I truly am fascinated by mathematicians. The thing is, they don't do it because it's necessary (unless they're engineers or teachers or trying to land a shuttle safely on the moon), but because they really do enjoy it. And this is a mystery to me. I've got pretty much everything figured out but this. I feel like somehow, there's gotta be a dopaminergic reward system for doing this stuff, set up in their brains... or else they'd use their intelligence for solving more immediate problems.

Listen if I didn't huff gas as a kid and drop out of school, I'd prolly be right here with y'all contemplating the elegance of this language and solving all kinds of shit for ya. But alas, I am fried, and so I shall only stare bewhildered while you work.

Thank you, and please do continue.
Scott Mayers
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Re: Occam's Razor vs. Principa Mathematica

Post by Scott Mayers »

That 360 pages is trivial given you are likely referencing an ABRIDGED version!

But to the reason for the degree of depth to which Russell intended was based upon the concept of PROOF, not merely some goal to point out how 1 + 1 = 2. The idea was to begin by using only the concept of sets and propositions without concern for its content to prove step by step a foundation of order out of chaos that needs NO MIND in principle to PROVE all logical relationships that can then be applied as a foundation for science.

The original effort begun with his "Principles of Mathematics" but he opted to start over with the help of Whitehead recognizing the depth needed was more than he realized. Thus the name he borrowed for the Latin of this title, "Principia Mathematica" and this was also to respect Isaac Newton's similar intent to understand the "logic" of nature. But this was also in hopes to find a foundational logic that can prove all of physical reality, not merely math.

"Mathematics" referred to anything that used a very strict logical set of arguments and proofs, not merely today's understanding of 'math' as a more distinct reference to the manipulation of numbers. ..and why Newton's name for a physic text was labeled as a apparently 'mathematical'. And "logic" is the actual generic topic of which modern 'math' is just a subset. Science is just the logic basing its premises on sensations as its input statements in arguments.

Because uncertainty about where to begin reasoning about Nature has always been conflicting when also used in social subjects that deal with things like politic and religion, Science became the mean of trying to understand things from only the senses as inputs to which anyone is presumed to be able to agree to.

Science is a 'top-down' approach to understanding nature and is akin to one beginning with some machine that one wants to understand by taking it apart from a whole. Step by step you take such 'machines' apart by observing and carefully noting everything as you do so. When you reach parts that are unable to be understood literally, you then guess at its 'logic' .....how the machine actually works. Then you test your logic by trying to reconstruct the 'machine' to see if its output matches to the original whole. If it WORKS, it gets accepted.

Logic is more formally the 'bottom up' processes of reasoning using real or assumed inputs to construct or reconstruct a 'machine' (or 'mechanical process'). Technically it actually includes science's top-down processing as well. We just separated logic proper into "formal" (deduction) versus "informal" (induction), where science is mainly based upon induction because the initial premises that go into logical constructs are not all 'apriori' or, at least, not able for all people to agree to which abstract ideas are sound.

Bertrand Russell was attempting to argue from the simplest elemental concepts that Nature itself would have had to use without any reliance on actual direct observations to see if you can prove everything physically real as based upon nothing but abstract concepts that have no intrinsic meaning. This ideal was what summarized the whole era labeled, "Modern" and "rationalist". "Postmodern" by its contrast references the stage from about the 1960s on that opted NOT to teach, learn and procede FOUNDATIONALLY (bottom-up) but in a CONTEMPORARY (top-down) style that emphasized 'abstracting' things into layers so that one can functionally learn science and technology by compartmentalizing the subjects in a way that doesn't require one to know everything. If you saw the actual 3 volumes of Principia Mathematica by Russell and Whitehead, you'd see why.

Examples: You can learn to program in a higher-order language without actually knowing how the lower layer mechanisms like the electronics or the machine language's of computers. You can learn to become a driver of a complex vehical like a jet airplane without requiring to know all the details of its particular engineering.
Age
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Re: Occam's Razor vs. Principa Mathematica

Post by Age »

promethean75 wrote: Mon Dec 27, 2021 9:43 pm Whenever my head gets too big from beating up philosophers at some forum, I find the nearest math thread so I can purge myself of excess pride and atone for being such a badass.
But you do not even accept all of the challenges provided to you, nor do you even answer all of the clarifying questions posed to you, in this philosophy forum, so this means that it is 'you', "promethean75", whoal is getting "beaten up" here.
promethean75 wrote: Mon Dec 27, 2021 9:43 pm I truly am fascinated by mathematicians. The thing is, they don't do it because it's necessary (unless they're engineers or teachers or trying to land a shuttle safely on the moon), but because they really do enjoy it. And this is a mystery to me. I've got pretty much everything figured out but this. I feel like somehow, there's gotta be a dopaminergic reward system for doing this stuff, set up in their brains... or else they'd use their intelligence for solving more immediate problems.

Listen if I didn't huff gas as a kid and drop out of school, I'd prolly be right here with y'all contemplating the elegance of this language and solving all kinds of shit for ya. But alas, I am fried, and so I shall only stare bewhildered while you work.

Thank you, and please do continue.
Age
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Re: Occam's Razor vs. Principa Mathematica

Post by Age »

roydop wrote: Wed Jun 09, 2021 2:45 pm The principle mathematical "proof" that 1+1=2 is 360 pages. Occam's Razor cuts that to: "One plus one equals two because we all believe that it does."
This is not exactly right, but what is exactly True is close enough to this, for now.
roydop wrote: Wed Jun 09, 2021 2:45 pm Anything that a 5 year old can understand should not require 360 pages concluding in

http://www.ivica-crnkovic.net/1+1=2.htm

to express this truth.

This looks like a programming language. A program designed as instruction for human consciousness.
Skepdick
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Re: Occam's Razor vs. Principa Mathematica

Post by Skepdick »

alan1000 wrote: Mon Dec 27, 2021 2:59 pm 1 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2
That's a piss-poor proof!

In particular this substitution is dubious: x + S(0) = S(x + 0)

If that is an acceptable substitution then here's an alernative...

1 + 1 = S(0) + S(0) = S (0 + 0) = S(0) = 1
promethean75
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Re: Occam's Razor vs. Principa Mathematica

Post by promethean75 »

"so this means that it is 'you', "promethean75", whoal is getting "beaten up" here."

Never happen, kid. I'm a third degree black belt of sophistry. I could be tag-teamed by Socrates AND Protagoras and I'd still win. Master of logos, pathos and ethos, I've read Aristotle's Rhetorics seven times, frontwards and backwards, and translated it into fourteen different languages.
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FlashDangerpants
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Re: Occam's Razor vs. Principa Mathematica

Post by FlashDangerpants »

Age wrote: Wed Dec 29, 2021 3:18 am
roydop wrote: Wed Jun 09, 2021 2:45 pm The principle mathematical "proof" that 1+1=2 is 360 pages. Occam's Razor cuts that to: "One plus one equals two because we all believe that it does."
This is not exactly right, but what is exactly True is close enough to this, for now.
Wait a minute .... are you pointing out that roydop hasn't accurately described what Occam's Razor is?



Chaps, we may have a christmas miracle on our hands here, Ken appears to have got something right.
Age
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Re: Occam's Razor vs. Principa Mathematica

Post by Age »

FlashDangerpants wrote: Fri Dec 31, 2021 8:10 pm
Age wrote: Wed Dec 29, 2021 3:18 am
roydop wrote: Wed Jun 09, 2021 2:45 pm The principle mathematical "proof" that 1+1=2 is 360 pages. Occam's Razor cuts that to: "One plus one equals two because we all believe that it does."
This is not exactly right, but what is exactly True is close enough to this, for now.
Wait a minute .... are you pointing out that roydop hasn't accurately described what Occam's Razor is?
NO, NOT AT ALL.

WHY would you ASSUME such a thing as this?
FlashDangerpants wrote: Fri Dec 31, 2021 8:10 pm
Chaps, we may have a christmas miracle on our hands here, Ken appears to have got something right.
But, SADLY, "flashdangerpants" was completely and utterly Wrong and WAY OFF, ONCE AGAIN.

So, ONCE MORE, I suggest NEVER ASSUMING ANY thing BEFORE you GAIN CLARITY FIRST.

Also, if you want to make a CLAIM like, "I have NEVER got ANY thing right", then PLEASE do NOT be TOO AFRAID to bring ANY of them up and POINT THEM OUT to the readers so that we ALL can have a LOOK AT your ACTUAL CLAIM, and NOT some IMAGINED ONES inside that head.

That is; If you are NOT TO SCARED and AFRAID to do so.

What has also been VERY CLEARLY NOTICED is your complete INABILITY to back up and support ANY of YOUR previous CLAIMS SO FAR, so there is NO reason to START ASSUMING you could or would THIS TIME.
Age
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Re: Occam's Razor vs. Principa Mathematica

Post by Age »

promethean75 wrote: Wed Dec 29, 2021 9:44 pm "so this means that it is 'you', "promethean75", whoal is getting "beaten up" here."

Never happen, kid.
LOL EACH time you are INCAPABLE TO or FAIL to accept the CHALLENGES posed to you here, or do NOT CLARIFY questions posed to you, 'you' are BEATEN UP "promethean75", which, by the way, has been EVERY time in relation to 'me'.
promethean75 wrote: Wed Dec 29, 2021 9:44 pm I'm a third degree black belt of sophistry. I could be tag-teamed by Socrates AND Protagoras and I'd still win. Master of logos, pathos and ethos, I've read Aristotle's Rhetorics seven times, frontwards and backwards, and translated it into fourteen different languages.
So what?

You have STILL FAILED to PROVE ANY CLAIM you have made so, which I have QUESTIONED and CHALLENGED you about.
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FlashDangerpants
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Re: Occam's Razor vs. Principa Mathematica

Post by FlashDangerpants »

Age wrote: Sat Jan 01, 2022 3:23 am
FlashDangerpants wrote: Fri Dec 31, 2021 8:10 pm
Age wrote: Wed Dec 29, 2021 3:18 am

This is not exactly right, but what is exactly True is close enough to this, for now.
Wait a minute .... are you pointing out that roydop hasn't accurately described what Occam's Razor is?
NO, NOT AT ALL.
Panic over. Everybody go about your business, normal service has been resumed.
Age
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Re: Occam's Razor vs. Principa Mathematica

Post by Age »

FlashDangerpants wrote: Sat Jan 01, 2022 2:54 pm
Age wrote: Sat Jan 01, 2022 3:23 am
FlashDangerpants wrote: Fri Dec 31, 2021 8:10 pm
Wait a minute .... are you pointing out that roydop hasn't accurately described what Occam's Razor is?
NO, NOT AT ALL.
Panic over. Everybody go about your business, normal service has been resumed.
So-called "normal service" just persists, as "flashdangerpants" continues to make ASSUMPTIONS, and be completely Wrong about them, be completely INCAPABLE to back up or PROVE its CLAIMS, and completely INCAPABLE to clarify things, but "flashdangerpants, does provide some GREAT EXAMPLES of what NOT to do in discussions.
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FlashDangerpants
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Re: Occam's Razor vs. Principa Mathematica

Post by FlashDangerpants »

Age wrote: Sun Jan 02, 2022 5:04 am "flashdangerpants" continues to make ASSUMPTIONS, and be completely Wrong about them
It's true. I assumed there was the outside possiblity of Ken being right on some matter.

I should be punished for this insolence, please say LOL, LOL, LOL at me, in that ever so effective way you have of doing things in threes. I will reform.
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