From 89de781affe045196b487755de67072608cff7f0 Mon Sep 17 00:00:00 2001 From: Thomas Walker Lynch Date: Wed, 26 Aug 2026 09:03:54 +0000 Subject: [PATCH] fixes ASCII art --- document/book/TM-2026.html | 147 +++++++++++++++++++------------------ 1 file changed, 76 insertions(+), 71 deletions(-) diff --git a/document/book/TM-2026.html b/document/book/TM-2026.html index 160edd9..780567e 100644 --- a/document/book/TM-2026.html +++ b/document/book/TM-2026.html @@ -392,64 +392,69 @@ - - Why the title of this book + + Establishing the book title - - Platonism -

Plato (427−347 BC) spoke of an entity with an independent existence that he called a Form. A Form is immutable, non-spatial, and non-temporal, and exists in a realm that has no location, called the Realm of Forms. Examples of Forms include mathematical properties (Equality, Circularity, Squareness, Unity, Doubleness), moral concepts (Justice, Courage, Piety, Temperance), aesthetic ideals (Beauty), physical archetypes (Tableness, Bedness, Treeness), and above them all the Good, which Plato held to be what makes the other Forms knowable. Plato then explains that knowledge is the conceptualization of Forms.

+ + The dialectic + + + Platonism -

- When people sense the material world, what they are sensing are the shadows from the Realm of Forms. Plato explains this with an allegory about a world where people are chained in a cave for their entire lives in a manner they only face a wall. Behind them a fire throws the shadows of objects onto the wall. Having never seen anything else, they mistake the shadows for reality itself. One of them is freed, is brought up into the light, and sees the objects themselves; and when he returns to tell the others, they do not believe him. - Plato, Republic VII 514a−520a. -

+

Plato (427−347 BC) spoke of an entity with an independent existence that he called a Form. A Form is immutable, non-spatial, and non-temporal, and exists in a realm that has no location, called the Realm of Forms. Examples of Forms include mathematical properties (Equality, Circularity, Squareness, Unity, Doubleness), moral concepts (Justice, Courage, Piety, Temperance), aesthetic ideals (Beauty), physical archetypes (Tableness, Bedness, Treeness), and above them all the Good, which Plato held to be what makes the other Forms knowable. Plato then explains that knowledge is the conceptualization of Forms.

-

As an illustration of the independent existence of Forms, consider what Plato says about geometers in the Republic. They draw a square and a diagonal, and then they argue about it. But they are not arguing about the figure they drew, and everyone participating in the discussion knows this to be the case. They are not asking whether that line, the one in the sand, is straight, nor whether that corner is square, for it plainly is not, and no conclusion they reach depends upon it. The drawn figure is used as an image of the Square itself, and what they say holds of the Square itself whether or not the drawing was made well, or made at all. The lines in the sand are merely the shadow of the Square being discussed. - Plato, Republic VI 510c−511a. The passage places mathematics on the third segment of the divided line, above opinion and below the account that would justify its hypotheses. Aristotle grants the same observation about mathematical practice and gives it the opposite explanation, at Metaphysics XIII 1078a17−31. -

+

+ When people sense the material world, what they are sensing are the shadows from the Realm of Forms. Plato explains this with an allegory about a world where people are chained in a cave for their entire lives in a manner they only face a wall. Behind them a fire throws the shadows of objects onto the wall. Having never seen anything else, they mistake the shadows for reality itself. One of them is freed, is brought up into the light, and sees the objects themselves; and when he returns to tell the others, they do not believe him. + Plato, Republic VII 514a−520a. +

-

When a geometer erases his drawing the theorem is untouched. He draws it again badly and the theorem is untouched. Every square that has ever been drawn could be wiped away and the theorem would still hold, and it held before any of them were drawn. Such is the non-temporal nature of Forms.

+

As an illustration of the independent existence of Forms, consider what Plato says about geometers in the Republic. They draw a square and a diagonal, and then they argue about it. But they are not arguing about the figure they drew, and everyone participating in the discussion knows this to be the case. They are not asking whether that line, the one in the sand, is straight, nor whether that corner is square, for it plainly is not, and no conclusion they reach depends upon it. The drawn figure is used as an image of the Square itself, and what they say holds of the Square itself whether or not the drawing was made well, or made at all. The lines in the sand are merely the shadow of the Square being discussed. + Plato, Republic VI 510c−511a. The passage places mathematics on the third segment of the divided line, above opinion and below the account that would justify its hypotheses. Aristotle grants the same observation about mathematical practice and gives it the opposite explanation, at Metaphysics XIII 1078a17−31. +

-

- The definition of a symbol given in section is Platonic in its structure. It begins with an unnamed math object that has required properties. The definition then states that we do not write the symbol directly, as it remains abstract. A representation convention is chosen for example, a Greek letter say, and what appears on a page is an instance of that representation. Hence the Form is the symbol, the shape of the shadow is the representation, and the shadow itself is an instance. -

+

When a geometer erases his drawing the theorem is untouched. He draws it again badly and the theorem is untouched. Every square that has ever been drawn could be wiped away and the theorem would still hold, and it held before any of them were drawn. Such is the non-temporal nature of Forms.

-

- If we were to rephrase the explanation of the geometers drawing squares in the sand in the language of this book, we would say there exists a math object distinct from other math objects. It is not written down directly, thus remains an abstraction. The shape the geometers recognize as a square is its representation. The knowledge of abstract concepts and their representations forms the common knowledge that is required for communication. The instance scratched into the sand is recognized as the representation and then facilitates the discussion of the concepts. -

+

+ The definition of a symbol given in section is Platonic in its structure. It begins with an unnamed math object that has required properties. The definition then states that we do not write the symbol directly, as it remains abstract. A representation convention is chosen for example, a Greek letter say, and what appears on a page is an instance of that representation. Hence the Form is the symbol, the shape of the shadow is the representation, and the shadow itself is an instance. +

-

- A subtle difference remains between the Symbol as a Form and the symbol of section . The structure is the same in both: an unwritten thing above, a representation, and many instances below. What differs is where the thing above is found. A Form is found in the Realm of Forms. The symbol of section is found among math objects. A math object, being anything a mathematician cares to name, includes representations and instances, the only requirements are that the symbols be distinct and can be represented. -

+

+ If we were to rephrase the explanation of the geometers drawing squares in the sand in the language of this book, we would say there exists a math object distinct from other math objects. It is not written down directly, thus remains an abstraction. The shape the geometers recognize as a square is its representation. The knowledge of abstract concepts and their representations forms the common knowledge that is required for communication. The instance scratched into the sand is recognized as the representation and then facilitates the discussion of the concepts. +

- +

+ A subtle difference remains between the Symbol as a Form and the symbol of section . The structure is the same in both: an unwritten thing above, a representation, and many instances below. What differs is where the thing above is found. A Form is found in the Realm of Forms. The symbol of section is found among math objects. A math object, being anything a mathematician cares to name, includes representations and instances, the only requirements are that the symbols be distinct and can be represented. +

- - Aristotelianism + -

The Realm of Forms is not anywhere, nothing done in this world bears upon what is true of it, and the man who has been up into the light returns with a report that can be checked against no other thing. Whether the report is believed depends upon the reputation of the man who came back with the testimony. That is the arrangement the priestess at Delphi works under, and it is the arrangement upon which every claim about an unseen realm hinges. This type of reasoning attracts those who believe in mysticism, and it repels those who seek grounding.

+ + Aristotelianism -

Aristotle (384−322 BC) arrived at Plato’s Academy when he was seventeen and Plato was sixty. Aristotle was the son of a physician, and had a pragmatic view. Where Plato was brilliant and intuitive, Aristotle was scholarly and methodical. The difference shows in what each took to be the starting point of an inquiry. Due to the passing of the centuries, not much is known about their personal dynamic at the Academy, though that Aristotle remained there for twenty years suggests that it was rich. - Will Buckingham, Douglas Burnham, Peter J. King, John Marenbon, Clive Hill, and Marcus Weeks, The Philosophy Book, Big Ideas Simply Explained (New York: DK, 2011), page 58. -

+

The Realm of Forms is not anywhere, nothing done in this world bears upon what is true of it, and the man who has been up into the light returns with a report that can be checked against no other thing. Whether the report is believed depends upon the reputation of the man who came back with the testimony. That is the arrangement the priestess at Delphi works under, and it is the arrangement upon which every claim about an unseen realm hinges. This type of reasoning attracts those who believe in mysticism, and it repels those who seek grounding.

-

Aristotle and Plato both rejected relativism, they both disagreed with Democritus’s theory of the atom, and they both held that the universe appeared to have some basic guiding principles. However, Aristotle rejected the Realm of Forms entirely. His objection was that it explains nothing: to account for the many circles in the world by positing one Circle elsewhere is not to clarify the situation, but rather it is to introduce yet another thing in need of explanation. To say that a drawn circle participates in the Circle is, in his words, to speak in poetical metaphors. - Aristotle, Metaphysics I.9 990b−991b, with the remark on poetical metaphors at 991a20−22, and again at XIII.4−5. Aristotle does use the word eidos, ordinarily translated as form, for the structure in virtue of which a thing is what it is. That structure is in the thing and nowhere else, and it is not Plato’s Form under another name. One English word is doing service for two concepts, and only the first is at issue here. - Take away the Realm of Forms, and there can be no shadows, nor a cave to be let out of.

+

Aristotle (384−322 BC) arrived at Plato’s Academy when he was seventeen and Plato was sixty. Aristotle was the son of a physician, and had a pragmatic view. Where Plato was brilliant and intuitive, Aristotle was scholarly and methodical. The difference shows in what each took to be the starting point of an inquiry. Due to the passing of the centuries, not much is known about their personal dynamic at the Academy, though that Aristotle remained there for twenty years suggests that it was rich. + Will Buckingham, Douglas Burnham, Peter J. King, John Marenbon, Clive Hill, and Marcus Weeks, The Philosophy Book, Big Ideas Simply Explained (New York: DK, 2011), page 58. +

-

For Plato, knowledge descends from on high: a person conceptualizes the Forms, and the senses report shadows that mislead as much as they inform. For Aristotle, knowledge ascends: observation of particulars is where an inquiry starts, and many observations lead to the deduction of what they have in common. - Aristotle, Posterior Analytics II.19, for knowledge beginning in perception and rising by induction to the principles. The Greek term for the operation is epagōgē, rendered as induction. -

+

Aristotle and Plato both rejected relativism, they both disagreed with Democritus’s theory of the atom, and they both held that the universe appeared to have some basic guiding principles. However, Aristotle rejected the Realm of Forms entirely. His objection was that it explains nothing: to account for the many circles in the world by positing one Circle elsewhere is not to clarify the situation, but rather it is to introduce yet another thing in need of explanation. To say that a drawn circle participates in the Circle is, in his words, to speak in poetical metaphors. + Aristotle, Metaphysics I.9 990b−991b, with the remark on poetical metaphors at 991a20−22, and again at XIII.4−5. Aristotle does use the word eidos, ordinarily translated as form, for the structure in virtue of which a thing is what it is. That structure is in the thing and nowhere else, and it is not Plato’s Form under another name. One English word is doing service for two concepts, and only the first is at issue here. + Take away the Realm of Forms, and there can be no shadows, nor a cave to be let out of.

+ +

For Plato, knowledge descends from on high: a person conceptualizes the Forms, and the senses report shadows that mislead as much as they inform. For Aristotle, knowledge ascends: observation of particulars is where an inquiry starts, and many observations lead to the deduction of what they have in common. + Aristotle, Posterior Analytics II.19, for knowledge beginning in perception and rising by induction to the principles. The Greek term for the operation is epagōgē, rendered as induction. +

-

After years of challenging Plato over the Realm of Forms, the debate ended when Plato passed away. Aristotle was not chosen to take his place to head the Academy. The chair went to Plato’s nephew Speusippus, a name few people are familiar with today, and the estate to family. - Speusippus, son of Plato’s sister Potone, headed the school for eight years; Diogenes Laertius, Lives IV.1. The will leaves the estate at Iphistiadae to a boy named Adeimantus, taken to be a young kinsman since Plato had a brother of that name, and lists a second estate at Eiresidae; Lives III.41−42. Headship of the school and title to the land were separate things, and neither came to Aristotle. The Academy grove itself was a public gymnasium, not Plato’s to bequeath, and Aristotle, a Stagirite and therefore a metic at Athens, could not have held Attic land in any case. - Upon leaving Aristotle dedicated his time to studying the natural world. He dissected, he collected, and he questioned fishermen and beekeepers about what they had seen; his account of the developing chick came from opening eggs on successive days. - Aristotle, Historia Animalium VI.3 561a for the chick; V.12 541b for the modified arm of the male octopus, which was thought a fable until confirmed in 1857. He was wrong about a great deal, holding the heart to be the seat of thought and the brain an organ for cooling the blood. Darwin, thanking William Ogle in 1882 for a translation of the Parts of Animals, wrote that Linnaeus and Cuvier had been his two gods, but that they were mere schoolboys compared to old Aristotle. Charles Darwin to William Ogle, 22 February 1882, in Francis Darwin, ed., The Life and Letters of Charles Darwin (London: John Murray, 1887), vol. 3, 252. - What he produced from it was a taxonomy: animals sorted by the features they share, the sorting answerable to the specimens and revised whenever a specimen refused its category.

+

After years of challenging Plato over the Realm of Forms, the debate ended when Plato passed away. Aristotle was not chosen to take his place to head the Academy. The chair went to Plato’s nephew Speusippus, a name few people are familiar with today, and the estate to family. + Speusippus, son of Plato’s sister Potone, headed the school for eight years; Diogenes Laertius, Lives IV.1. The will leaves the estate at Iphistiadae to a boy named Adeimantus, taken to be a young kinsman since Plato had a brother of that name, and lists a second estate at Eiresidae; Lives III.41−42. Headship of the school and title to the land were separate things, and neither came to Aristotle. The Academy grove itself was a public gymnasium, not Plato’s to bequeath, and Aristotle, a Stagirite and therefore a metic at Athens, could not have held Attic land in any case. + Upon leaving Aristotle dedicated his time to studying the natural world. He dissected, he collected, and he questioned fishermen and beekeepers about what they had seen; his account of the developing chick came from opening eggs on successive days. + Aristotle, Historia Animalium VI.3 561a for the chick; V.12 541b for the modified arm of the male octopus, which was thought a fable until confirmed in 1857. He was wrong about a great deal, holding the heart to be the seat of thought and the brain an organ for cooling the blood. Darwin, thanking William Ogle in 1882 for a translation of the Parts of Animals, wrote that Linnaeus and Cuvier had been his two gods, but that they were mere schoolboys compared to old Aristotle. Charles Darwin to William Ogle, 22 February 1882, in Francis Darwin, ed., The Life and Letters of Charles Darwin (London: John Murray, 1887), vol. 3, 252. + What he produced from it was a taxonomy: animals sorted by the features they share, the sorting answerable to the specimens and revised whenever a specimen refused its category.

-

This perspective, that knowledge comes from the categorization of observations of nature rather than, as Plato taught, from the conceptualization of Forms, is the essence of Naturalism. There are two modern definitions for Naturalism. One definition is that of a philosophy that nothing exists outside of nature. This philosophy is generally taken to deny the existence of any abstraction standing apart from what could be observed, including that of the Realm of Forms. The other definition is that of a study of nature, that of a wildlife biologist.

+

This perspective, that knowledge comes from the categorization of observations of nature rather than, as Plato taught, from the conceptualization of Forms, is the essence of Naturalism. There are two modern definitions for Naturalism. One definition is that of a philosophy that nothing exists outside of nature. This philosophy is generally taken to deny the existence of any abstraction standing apart from what could be observed, including that of the Realm of Forms. The other definition is that of a study of nature, that of a wildlife biologist.

+ @@ -2461,22 +2466,22 @@ 
-              [ State Transition Table ]
-              +-------------------------------------------+
-              current state |  def   q0    q1    q2    q3    q4    q5   |
-              |         ●     ○     ○     ○     ○     ○   |
-              halt |        [/]   [/]   [/]   [/]   [/]   [/]  |
-              +-------------------------------------------+
-              def |  ( )   ( )   ( )   ( )   ( )   ( )   ( )  |
-              ● s0   |  ( )   ( )   ( )   ( )   ( )   ( )   ( )  |
-              ○ s1   |  ( )   ( )   ( )   ( )   ( )   ( )   ( )  |
-              ○ s2   |  ( )   ( )   ( )   ( )   ( )   ( )   ( )  |
-              ○ s3   |  ( )   ( )   ( )   ( )   ( )   ( )   ( )  |
-              ○ s4   |  ( )   ( )   ( )   ( )   ( )   ( )   ( )  |
-              |                                           |
-              destination state |        ( )   ( )   ( )   ( )   ( )   ( )  |
-              +-------------------------------------------+
-
+ [ State Transition Table ] + +-------------------------------------------+ + current state | def q0 q1 q2 q3 q4 q5 | + | ● ○ ○ ○ ○ ○ | + halt | [/] [/] [/] [/] [/] [/] | + +-------------------------------------------+ + def | ( ) ( ) ( ) ( ) ( ) ( ) ( ) | + ● s0 | ( ) ( ) ( ) ( ) ( ) ( ) ( ) | + ○ s1 | ( ) ( ) ( ) ( ) ( ) ( ) ( ) | + ○ s2 | ( ) ( ) ( ) ( ) ( ) ( ) ( ) | + ○ s3 | ( ) ( ) ( ) ( ) ( ) ( ) ( ) | + ○ s4 | ( ) ( ) ( ) ( ) ( ) ( ) ( ) | + | | + destination state | ( ) ( ) ( ) ( ) ( ) ( ) | + +-------------------------------------------+ + Figure . The state transition panel, with state indicators and halt switches @@ -2487,15 +2492,15 @@ [ Instruction ] +-----------------------------------+ | q0 q1 q2 q3 q4 q5 | - Src | ( ) ( ) ( ) ( ) ( ) ( ) | + Src | ( ) ( ) ( ) ( ) ( ) ( ) | +-----------------------------------+ - right | ( ) ( ) ( ) ( ) ( ) ( ) | - left | ( ) ( ) ( ) ( ) ( ) ( ) | - read('s') | ( ) ( ) ( ) ( ) ( ) ( ) | - read('d') | ( ) ( ) ( ) ( ) ( ) ( ) | - write('s')| ( ) ( ) ( ) ( ) ( ) ( ) | - write('d')| ( ) ( ) ( ) ( ) ( ) ( ) | - write('σ')| ( ) ( ) ( ) ( ) ( ) ( ) | + right | ( ) ( ) ( ) ( ) ( ) ( ) | + left | ( ) ( ) ( ) ( ) ( ) ( ) | + read('s') | ( ) ( ) ( ) ( ) ( ) ( ) | + read('d') | ( ) ( ) ( ) ( ) ( ) ( ) | + write('s')| ( ) ( ) ( ) ( ) ( ) ( ) | + write('d')| ( ) ( ) ( ) ( ) ( ) ( ) | + write('σ')| ( ) ( ) ( ) ( ) ( ) ( ) | +-----------------------------------+ Figure . The instruction panel, one patch row per instruction and one column per state @@ -2508,13 +2513,13 @@ [ Sigma Select for write('σ') ] +-----------------------------------+ | q0 q1 q2 q3 q4 q5 | - Src | ( ) ( ) ( ) ( ) ( ) ( ) | + Src | ( ) ( ) ( ) ( ) ( ) ( ) | +-----------------------------------+ - s0 | ( ) ( ) ( ) ( ) ( ) ( ) | - s1 | ( ) ( ) ( ) ( ) ( ) ( ) | - s2 | ( ) ( ) ( ) ( ) ( ) ( ) | - s3 | ( ) ( ) ( ) ( ) ( ) ( ) | - s4 | ( ) ( ) ( ) ( ) ( ) ( ) | + s0 | ( ) ( ) ( ) ( ) ( ) ( ) | + s1 | ( ) ( ) ( ) ( ) ( ) ( ) | + s2 | ( ) ( ) ( ) ( ) ( ) ( ) | + s3 | ( ) ( ) ( ) ( ) ( ) ( ) | + s4 | ( ) ( ) ( ) ( ) ( ) ( ) | +-----------------------------------+ Figure . The sigma select panel, choosing the symbol written by write(σ) -- 2.20.1