Order without orientation (New version --- in progress)
order_without_orientation.docx | |
File Size: | 163 kb |
File Type: | docx |
order_without_orientation.pdf | |
File Size: | 587 kb |
File Type: |
Last revised 6/15/2011, uploaded 6/7/2011
Order without orientation (Old, NG version)
order_without_orientation.docx | |
File Size: | 36 kb |
File Type: | docx |
order_without_orientation.pdf | |
File Size: | 522 kb |
File Type: |
Uploaded 3/22/2011, last revised 3/27/2011
Element is relation: Postscript
element_is_relation.docx | |
File Size: | 26 kb |
File Type: | docx |
element_is_relation.pdf | |
File Size: | 419 kb |
File Type: |
Uploaded 3/25/2011, last revised 3/28/2011
Comments
(6/7/2011, revised slightly 6/15/2011)
I just uploaded a new version of "Order without orientation." This is still a work in progress. But, I will put it here just in case today. (I will fly by plane tomorrow, 6/8/2011. I somehow thought of a possibility of dying by accident before finishing this essay.)
The new version is a considerable revision, which requires a revision of the "Element is relation." That is, the old version of both "Order without relation" and "Element is relation" will be eventually integrated into the complete version of the current version.
(The old versions are kept here for a while, until the current version is finished and uploaded.)
I just uploaded a new version of "Order without orientation." This is still a work in progress. But, I will put it here just in case today. (I will fly by plane tomorrow, 6/8/2011. I somehow thought of a possibility of dying by accident before finishing this essay.)
The new version is a considerable revision, which requires a revision of the "Element is relation." That is, the old version of both "Order without relation" and "Element is relation" will be eventually integrated into the complete version of the current version.
(The old versions are kept here for a while, until the current version is finished and uploaded.)
[From the paper] (3/22/2011, last revised 3/28/2011)
Two mathematical structures, <Q+, ÷> (where Q+ is the set of positive rationals and ÷ is division) and <Z, -> (where Z is the set of integers and - is subtraction) are models of the same theory. This theory seems to exhibit a peculiar property of introducing an "orientation-neutral linear order" to the structure.
This essay is basically just a study memo, which I put online to ask for other's help concerning the definition of this theory, and for response to my thoughts on this peculiar property. In this memo, I call this theory ying theory, and call each model of this theory a ying, following the convention of mathematical discourse. But, based on my past experience, I'm almost 100% sure that this is a known theory, if it is definable at all. On the other hand, I'm wondering if my thoughts on its peculiar property may be novel, and may deserve attention of those who are interested in the philosophy of mathematics (and language, which for me is the same thing ultimately).
The postscript, "Element is relation," gives another theorem of ying theory, in comparison to Cayley's theorem of group theory. (I uploaded this postscript twice on 3/25/2011 because the first version contained a serious blunder on the definition of Cayley's theorem. I apologize if this caused any inconvenience to anyone. Minor spelling and typing errors are fixed on 3/28/2011.)
Two mathematical structures, <Q+, ÷> (where Q+ is the set of positive rationals and ÷ is division) and <Z, -> (where Z is the set of integers and - is subtraction) are models of the same theory. This theory seems to exhibit a peculiar property of introducing an "orientation-neutral linear order" to the structure.
This essay is basically just a study memo, which I put online to ask for other's help concerning the definition of this theory, and for response to my thoughts on this peculiar property. In this memo, I call this theory ying theory, and call each model of this theory a ying, following the convention of mathematical discourse. But, based on my past experience, I'm almost 100% sure that this is a known theory, if it is definable at all. On the other hand, I'm wondering if my thoughts on its peculiar property may be novel, and may deserve attention of those who are interested in the philosophy of mathematics (and language, which for me is the same thing ultimately).
The postscript, "Element is relation," gives another theorem of ying theory, in comparison to Cayley's theorem of group theory. (I uploaded this postscript twice on 3/25/2011 because the first version contained a serious blunder on the definition of Cayley's theorem. I apologize if this caused any inconvenience to anyone. Minor spelling and typing errors are fixed on 3/28/2011.)
(3/22/2011, last revised 3/25/2011)In the main essay, I adopt the common mathematical convention of calling what I call (in "Two ways of identification") a structure or a simple universal a theory and what I call (ditto) a model of a structure or a complex particular a structure. I hope that this does not confuse anyone who reads this after the "Two ways of identification."
(3/23/2011, last revised 3/27/2011)
The version I uploaded 3/22/2011 (the very first version) contained several typos and significant errors, one even in definition. Hopefully this new version I just uploaded (on 3/23) fixes the main problems, getting the definition and proofs right, at least. Obviously I need to investigate theorems more. I apologize if someone already took time to read it and tried to see if theorems really follow from the definition.
(And I did this twice today, 3/23/2011. I made yet another emendation to the definition. Apology again.)
(And a few more minor changes, concerning terminology definition, on 3/24/2011.)
(An improvement is made on the definition of the ying theory, and a very minor typo is fixed there, too, on 3/25/2011.)
(Another serious error was found in the definition, and was fixed. 3/27/2011)
The version I uploaded 3/22/2011 (the very first version) contained several typos and significant errors, one even in definition. Hopefully this new version I just uploaded (on 3/23) fixes the main problems, getting the definition and proofs right, at least. Obviously I need to investigate theorems more. I apologize if someone already took time to read it and tried to see if theorems really follow from the definition.
(And I did this twice today, 3/23/2011. I made yet another emendation to the definition. Apology again.)
(And a few more minor changes, concerning terminology definition, on 3/24/2011.)
(An improvement is made on the definition of the ying theory, and a very minor typo is fixed there, too, on 3/25/2011.)
(Another serious error was found in the definition, and was fixed. 3/27/2011)