Popular in Arts and Humanities

Arts and Humanities

Freedom’s Dominion

Vanderbilt University Jefferson Cowie

Arts and Humanities

Titus Quinctius Flamininus’ “Italian triumph”

McGill University Michael Fronda

Arts and Humanities

Lincoln and the Jews

Brandeis University Jonathan Sarna

Arts and Humanities

Amputations and Convalescence

University of Western Australia Emily Brink

Art

Drawing on the past: Palladio, his precursors and knowledge of ancient architecture

University of Birmingham David Hemsoll

Art

Language, Sound, and Image in Wallace Berman’s Aleph, 1956–1966

University of San Diego Chelsea Behle Fralick

Conceptual Harmonies

View more insights in Arts and Humanities - Philosophy

View more figures from University of Sydney

View the page of Paul Redding

Read the Study

Next in Arts and Humanities

Does the United States Need a Climate Refugee Policy?

Maria Cristina Garcia | Cornell University

The European Reformation

Euan Cameron | Columbia University

Paul Redding emphasizes Hegel’s focus on Pythagorean ratios, Platonic reason, and Aristotle’s geometric logic in the context of modern mathematics.

Image courtesy of interviewee. January 23, 2024

Log-in or Sign-up to Faculti
Currently viewing this subject insight as a guest. You have insight(s) remaining for this month. Login to view 8000+ figures on the platform.

Generating Workspace...

Export as PDF

The speaker discusses their interest in the Greek influences on Hegel’s logic, particularly the relationship between Aristotle's syllogistic logic and Plato's philosophical ideas as found in the "Timaeus." They explore the notion that Aristotle simplified Plato’s concepts, especially in relation to mathematics and geometry. The speaker points out the historical context of Greek mathematics, focusing on the impact of figures like Oxis of Tus and the Pythagorean music theory on later logic and geometry.

They highlight how Aristotle's syllogisms, concerned with logical structures, may have been influenced by geometric proofs, where diagrams and visual representations play a significant role. The speaker argues that recent scholarship has largely overlooked the musical and Pythagorean dimensions of Hegel's logic, which they believe deserves more attention.

They elaborate on how Hegel saw Aristotle's logic as needing a broader context that includes Platonic and Pythagorean influences, linking this to modern developments in mathematics, particularly projective geometry and its connections to music theory. The speaker mentions the importance of figures like Godfrey Leibniz, who contributed to logic and geometry in a way that resonates with Hegel's ideas. The overarching goal is to reintegrate Hegel into the discussion of modern logic, emphasizing his relevance and influence on contemporary mathematical thought.

Language:

Export as PDF

X-TIMESTAMP-MAP=LOCAL:00:00:00.000,MPEGTS:90000

1
00:00:00.025 --> 00:00:02.165
So I was primarily interested in,

2
00:00:02.185 --> 00:00:04.605
the Greek influences on Hegel's logic.

3
00:00:04.675 --> 00:00:07.205
I've been working on Hagel for too many years now,

4
00:00:07.785 --> 00:00:09.125
and I became interested in

5
00:00:09.125 --> 00:00:10.965
what is now known about the origins

6
00:00:10.965 --> 00:00:13.005
of Aristotle's Sistic logic,

7
00:00:13.625 --> 00:00:14.965
and in particular its relation

8
00:00:14.965 --> 00:00:16.965
to mathematics As it was practiced in that period,

9
00:00:18.895 --> 00:00:20.335
I was motivated by a claim

10
00:00:20.335 --> 00:00:22.375
that Hagel makes in his lectures on the history

11
00:00:22.375 --> 00:00:24.815
of philosophy, where he suggests

12
00:00:24.815 --> 00:00:28.375
that Aristotle's logic had been derived from the account

13
00:00:28.375 --> 00:00:30.135
of the universe that,

14
00:00:30.945 --> 00:00:34.455
Plato gives in this peculiar dialogue called the Timaeus.

15
00:00:34.515 --> 00:00:37.415
You know, the idea was that Aristotle derives his account

16
00:00:37.755 --> 00:00:40.255
of the logistic structure of inferences

17
00:00:40.355 --> 00:00:42.815
and so forth from this account and that,

18
00:00:42.835 --> 00:00:45.815
but that Aristo had simplified Plato's account,

19
00:00:46.275 --> 00:00:49.375
and Hagel says quite mysteriously he had reduced

20
00:00:50.005 --> 00:00:52.615
Plato's two means to one.

21
00:00:53.195 --> 00:00:55.255
Uh, so the question is, well, what are these means

22
00:00:55.325 --> 00:00:56.455
that he's talking about?

23
00:00:58.685 --> 00:01:01.545
Um, I think it became, pretty obvious that what,

24
00:01:01.575 --> 00:01:04.145
what he was appealing to were the three means

25
00:01:04.245 --> 00:01:08.225
or middle terms of these proportions that found in

26
00:01:08.865 --> 00:01:10.185
Pythagorean Music theory.

27
00:01:11.445 --> 00:01:14.425
Um, and that is actually structuring Plato's account

28
00:01:14.485 --> 00:01:17.305
of the universe that's given in the time maus.

29
00:01:18.015 --> 00:01:19.625
Luckily for me, there's been a lot

30
00:01:19.625 --> 00:01:22.145
of work done over the last 50 years

31
00:01:22.285 --> 00:01:24.225
or so on the history of Greek mathematics,

32
00:01:24.725 --> 00:01:26.025
and in particular a lot

33
00:01:26.025 --> 00:01:28.505
of really good work on Pythagorean music theory.

34
00:01:29.205 --> 00:01:32.185
So I set out to examine the, the intelligibility

35
00:01:32.325 --> 00:01:33.505
and the plausibility

36
00:01:34.205 --> 00:01:37.745
of hagel sketches right against what we now know.

37
00:01:39.845 --> 00:01:42.145
So many people have comment commented over the years

38
00:01:42.175 --> 00:01:44.625
that the influence, on the influence of,

39
00:01:44.915 --> 00:01:47.625
Greek geometry on Aristotle's logic,

40
00:01:47.965 --> 00:01:49.545
and that seems pretty plausible.

41
00:01:49.885 --> 00:01:52.545
We know a lot of the geometric theorems

42
00:01:52.545 --> 00:01:56.025
that ended up in Euclid's, elements, the great work

43
00:01:56.055 --> 00:01:59.465
that came from, you know, after Plato

44
00:01:59.465 --> 00:02:01.585
and Aristotle's time, we know that a lot

45
00:02:01.585 --> 00:02:04.305
of those had derived from the mathematics being practiced in

46
00:02:04.325 --> 00:02:05.985
Athens in the first half

47
00:02:05.985 --> 00:02:09.745
of the fourth century when Plato established his academy.

48
00:02:10.965 --> 00:02:13.705
And of course, the idea of proof from axioms

49
00:02:13.705 --> 00:02:15.745
and definitions to theories is central

50
00:02:15.845 --> 00:02:17.065
to Euclidean geometry.

51
00:02:17.205 --> 00:02:21.865
So Aristotle's logic concerns syllogisms in which pairs of

52
00:02:22.545 --> 00:02:25.265
premises, like two statements lead to

53
00:02:25.365 --> 00:02:26.705
by necessity, a conclusion.

54
00:02:27.525 --> 00:02:30.905
And it seems natural that he would've in some way been drawn

55
00:02:30.925 --> 00:02:32.505
to this, existing idea

56
00:02:32.565 --> 00:02:35.265
of the proof structures in geometry.

57
00:02:36.595 --> 00:02:38.535
Uh, together with this many have conjecture

58
00:02:38.535 --> 00:02:40.895
that Aristotle had actually employed diagrams

59
00:02:40.915 --> 00:02:43.775
or did teach logic, in a similar way

60
00:02:43.775 --> 00:02:45.855
to the way in which diagrams are employed in

61
00:02:45.855 --> 00:02:46.975
geometric proofs.

62
00:02:47.355 --> 00:02:49.375
So all that seemed pretty uncontroversial,

63
00:02:51.085 --> 00:02:53.905
but it was what what takes us beyond that

64
00:02:53.905 --> 00:02:55.065
to me, which was interesting.

65
00:02:55.715 --> 00:02:58.385
Plato had founded his academy, which is sort

66
00:02:58.385 --> 00:03:02.685
of like the first university, if you like, around 3 87,

67
00:03:03.225 --> 00:03:06.125
you know, before the common era BC

68
00:03:06.745 --> 00:03:07.885
and Aristotle was thought

69
00:03:07.885 --> 00:03:10.845
to have joined at about 20 years later as a teenager.

70
00:03:10.985 --> 00:03:13.565
So he was considerably, you know, Plato's Junior

71
00:03:14.555 --> 00:03:16.485
Aristotle has been particularly described

72
00:03:16.485 --> 00:03:20.565
as having been influenced by a mathematician named

73
00:03:21.075 --> 00:03:22.165
Oxis of Tus.

74
00:03:22.925 --> 00:03:25.645
A Oxis was a little older than Aristotle,

75
00:03:25.985 --> 00:03:28.525
but he apparently joined the Academy around the same time,

76
00:03:29.385 --> 00:03:32.085
and he's one of the really, sort of big contributors

77
00:03:32.085 --> 00:03:35.325
to Euclidean geometry in particular.

78
00:03:36.265 --> 00:03:39.205
Um, Oxis is thought to have introduced

79
00:03:40.155 --> 00:03:44.045
into geometry, into Euclid's elements, really,

80
00:03:44.225 --> 00:03:47.285
the contents of this particular book, book five, which is

81
00:03:47.285 --> 00:03:49.165
to do with proportions and ratios,

82
00:03:49.705 --> 00:03:50.925
and this is seen to be the sort

83
00:03:50.925 --> 00:03:55.525
of major move which established geometry as a type

84
00:03:55.525 --> 00:03:58.965
of autonomous discipline, autonomous from arithmetic,

85
00:03:59.175 --> 00:04:00.485
which the Aris, which the,

86
00:04:00.485 --> 00:04:02.045
which the Greeks weren't very good at.

87
00:04:02.145 --> 00:04:03.725
The Greek arithmetic more

88
00:04:03.725 --> 00:04:06.005
or less came from, Mesopotamia,

89
00:04:06.025 --> 00:04:08.085
and they never really developed it, so they had

90
00:04:08.085 --> 00:04:09.725
to break away from arithmetic

91
00:04:09.825 --> 00:04:12.645
to develop geometry in the way that they did.

92
00:04:12.735 --> 00:04:16.445
There was also another problem about, the relation

93
00:04:16.445 --> 00:04:17.685
of geometry and arithmetic,

94
00:04:17.685 --> 00:04:22.045
and that it was discovered that, ratios

95
00:04:22.145 --> 00:04:24.605
of line lengths such the sort of things

96
00:04:24.605 --> 00:04:26.325
that geometry is interested in, right?

97
00:04:26.325 --> 00:04:28.365
Like the ratio between the side

98
00:04:28.365 --> 00:04:29.725
and the diagonal of a square

99
00:04:30.695 --> 00:04:34.085
could actually couldn't be represented by ratios of numbers.

100
00:04:35.305 --> 00:04:39.805
Um, in modern terms, would would say that the ratio

101
00:04:40.365 --> 00:04:43.485
re required irrational numbers, right square root of two,

102
00:04:43.825 --> 00:04:45.885
but the Pythagorean simply didn't have that concept

103
00:04:45.945 --> 00:04:48.645
of number, so there was that problem.

104
00:04:49.545 --> 00:04:51.325
So a different approach was necessary,

105
00:04:51.345 --> 00:04:53.565
and Oxis is seen as the key person there.

106
00:04:54.145 --> 00:04:56.925
Now, Plato would've been in his early sixties by the time

107
00:04:56.925 --> 00:04:58.485
of these sort of discoveries,

108
00:04:59.025 --> 00:05:00.285
and his mathematics seems

109
00:05:00.285 --> 00:05:02.205
to have been much more dependent on the approach

110
00:05:02.205 --> 00:05:03.725
of earlier Pythagorean.

111
00:05:04.265 --> 00:05:07.325
Now, people such as, there's a character named Phil La

112
00:05:07.325 --> 00:05:11.805
of Croton Russ, roughly a contemporary of Socrates

113
00:05:11.825 --> 00:05:13.245
and age,

114
00:05:13.305 --> 00:05:18.245
and a person called Tis of Toum, who was roughly Plato's,

115
00:05:18.305 --> 00:05:22.765
age was supposedly a, a friend of Plato as well.

116
00:05:23.395 --> 00:05:26.045
They were the major music theorists of the time.

117
00:05:27.235 --> 00:05:31.855
This Pythagorean tradition with its musical references seems

118
00:05:31.855 --> 00:05:35.935
to be largely eclipsed by the geometry of Oxis by the time

119
00:05:35.935 --> 00:05:39.415
that developed, however, it persisted more

120
00:05:39.415 --> 00:05:41.575
or less in the background of Euclidean geometry

121
00:05:41.575 --> 00:05:44.295
and in other aspects of geometry at the time,

122
00:05:45.275 --> 00:05:49.055
and it became linked to later geometric discoveries.

123
00:05:49.235 --> 00:05:52.735
Um, the one I'm interested in is that discovery of the,

124
00:05:52.735 --> 00:05:54.575
so-called Connic sections.

125
00:05:54.635 --> 00:05:58.685
If you, if you think they're figures like circles, ellipses,

126
00:05:58.795 --> 00:06:01.245
parabolas and hyperbola, they're all linked up.

127
00:06:02.025 --> 00:06:04.845
The Greeks used to think of them as being,

128
00:06:04.895 --> 00:06:08.285
you'd get them by sectioning a cone in on different angles.

129
00:06:09.025 --> 00:06:11.765
So the Pythagorean tradition didn't entirely disappear,

130
00:06:12.225 --> 00:06:13.325
and in fact, it seems

131
00:06:13.325 --> 00:06:16.365
to have been revived in certain developments in modern

132
00:06:16.365 --> 00:06:20.165
mathematics that I think Hagel was, interested in,

133
00:06:20.465 --> 00:06:21.485
in Plato's time.

134
00:06:21.535 --> 00:06:25.205
Mayas the fictional character time Mayas who's it's thought

135
00:06:25.205 --> 00:06:28.965
that he may be, based on Akes or,

136
00:06:29.545 --> 00:06:31.125
or, you know, someone else around that time

137
00:06:32.225 --> 00:06:35.005
to Mayas describes the way the cosmos was put together

138
00:06:35.225 --> 00:06:37.845
by the great artifice, you know, sort of Plato's God

139
00:06:38.425 --> 00:06:41.245
as being held together by what Plato described

140
00:06:41.265 --> 00:06:42.845
as the most beautiful bond.

141
00:06:44.865 --> 00:06:46.805
Now that the structuring of ratios

142
00:06:46.805 --> 00:06:49.725
and proportions in this account was taken from Pythagorean.

143
00:06:49.725 --> 00:06:51.085
Music theory, as I mentioned,

144
00:06:51.945 --> 00:06:53.885
led into a tradition called the Music

145
00:06:53.945 --> 00:06:55.405
of the Spheres tradition.

146
00:06:56.465 --> 00:06:59.605
Now, in a seemingly sensible, people always dismiss this.

147
00:06:59.715 --> 00:07:02.365
Aristotle certainly dismissed it, the idea

148
00:07:02.365 --> 00:07:05.965
that the planets give up heavenly sort of notes

149
00:07:05.965 --> 00:07:07.325
that are all harmonious, right?

150
00:07:07.915 --> 00:07:09.925
Aristotle dismissed it as a beer metaphor.

151
00:07:10.755 --> 00:07:13.165
This tradition persisted up to the Renaissance,

152
00:07:13.625 --> 00:07:15.805
and in fact, later it was taken seriously

153
00:07:15.825 --> 00:07:18.445
by the great German astronomer, Johannes Kepper.

154
00:07:19.385 --> 00:07:22.285
Um, Kepper was actually from Hagels hometown,

155
00:07:22.285 --> 00:07:25.685
and Hagel was very attracted to Kepler's thinking on this.

156
00:07:26.625 --> 00:07:29.485
And Hagel thought that Kepler was clearly onto something

157
00:07:29.485 --> 00:07:30.685
with this, sort

158
00:07:30.685 --> 00:07:33.765
of platonic Pythagorean approach to cosmology.

159
00:07:34.825 --> 00:07:36.605
And Hagel was roundly criticized

160
00:07:36.705 --> 00:07:38.125
for buying into this tradition.

161
00:07:38.905 --> 00:07:43.715
But I think that that hagel took this in a more logical than

162
00:07:43.715 --> 00:07:45.235
a more cosmological sense, right?

163
00:07:45.235 --> 00:07:48.835
He was thinking of the structuring of Kepler's thinking

164
00:07:50.215 --> 00:07:53.475
it had something to do with the way that logic, the,

165
00:07:53.535 --> 00:07:55.155
the logic of the world, if you like,

166
00:07:55.175 --> 00:07:57.435
is captured in our scientific explanations.

167
00:07:59.055 --> 00:08:01.155
So his support of Plato's logic

168
00:08:01.225 --> 00:08:05.075
with its musical means over Aristotle's more reduced

169
00:08:05.475 --> 00:08:09.715
Euclidean form was meant to be, more adequate

170
00:08:09.735 --> 00:08:11.995
to this type of, modern astronomy.

171
00:08:12.115 --> 00:08:14.795
I think what struck me about Hegel's claim

172
00:08:15.855 --> 00:08:18.725
about Aristotle's logic as deriving from

173
00:08:19.515 --> 00:08:23.445
Plato's musical type of logic, is the fact

174
00:08:23.445 --> 00:08:26.365
that about a hundred years ago in the 1930s,

175
00:08:26.685 --> 00:08:31.565
a classicist named Benedict Anderson, pointed out

176
00:08:32.075 --> 00:08:35.885
that the vocabulary that Aristotle introduces for his logic,

177
00:08:36.025 --> 00:08:37.565
the specialist vocabulary,

178
00:08:38.065 --> 00:08:40.285
had actually been taken from the vocabulary

179
00:08:40.505 --> 00:08:42.245
of Pythagoria and music theory.

180
00:08:44.055 --> 00:08:47.035
Now, later, another historian of Greek mathematics,

181
00:08:47.315 --> 00:08:51.115
a a Russian named Abo also looked at the way

182
00:08:51.115 --> 00:08:54.755
that Pythagorean music had been brought into the geometric

183
00:08:54.805 --> 00:08:57.925
conceptions of ratios, the sort of important stuff

184
00:08:57.925 --> 00:09:01.285
that ox has developed from my perspective,

185
00:09:01.335 --> 00:09:03.645
these were the sorts of links that har that,

186
00:09:03.645 --> 00:09:06.925
that Hagel had been exploring in some sort of way

187
00:09:07.355 --> 00:09:08.965
between the lines of his account.

188
00:09:10.585 --> 00:09:12.805
So Hagel, I came to believe thought

189
00:09:12.805 --> 00:09:15.005
that a logic like Aristotle's

190
00:09:15.615 --> 00:09:18.325
model predominantly on Euclidean geometry needed

191
00:09:18.325 --> 00:09:21.685
to be supplemented by context concepts

192
00:09:21.685 --> 00:09:24.045
that drew on this non-Euclidean tradition

193
00:09:24.635 --> 00:09:27.885
with its earlier pla, platonic and Pythagorean roots.

194
00:09:29.205 --> 00:09:31.405
Moreover, I believe that he thought

195
00:09:31.405 --> 00:09:34.565
that certain modern trains of thought had also

196
00:09:35.135 --> 00:09:36.565
drawn upon this tradition.

197
00:09:38.155 --> 00:09:41.335
Um, in the book, I argue that I think that he was right.

198
00:09:42.075 --> 00:09:44.815
The mathematical approach to geometry that's now known

199
00:09:44.835 --> 00:09:46.095
as projective geometry,

200
00:09:46.345 --> 00:09:49.375
which was a huge development in the 19th century,

201
00:09:50.455 --> 00:09:54.855
actually had conceptual links with Greek musical theory,

202
00:09:55.115 --> 00:09:57.535
via its links to the theory of comic sections.

203
00:09:59.885 --> 00:10:02.865
Now, actually, this link is suggested by the fact that one

204
00:10:02.865 --> 00:10:04.225
of the central proportions

205
00:10:04.225 --> 00:10:08.025
of projective geometry became called, became known, known

206
00:10:08.045 --> 00:10:10.905
as the harmonic cross ratio.

207
00:10:12.085 --> 00:10:13.545
Um, and it goes back to

208
00:10:13.545 --> 00:10:16.305
what the Greek astronomer apollonius working

209
00:10:16.435 --> 00:10:17.905
after the time of Euclid

210
00:10:18.045 --> 00:10:20.105
and drawing on some of these musical ideas.

211
00:10:20.965 --> 00:10:24.465
Um, in, in his work on the Connic sections had described

212
00:10:24.465 --> 00:10:28.905
as the harmonic division of a line, a harmonic division

213
00:10:28.905 --> 00:10:32.025
of a line divided it into four points

214
00:10:32.935 --> 00:10:34.745
that could be ara, and then,

215
00:10:34.765 --> 00:10:37.465
and assigned numbers were assigned to those points

216
00:10:37.895 --> 00:10:40.505
that could be arranged in a type of double ratio.

217
00:10:41.925 --> 00:10:44.345
Now, this double ratio, as I argue in the book,

218
00:10:44.805 --> 00:10:48.305
is in fact a generalization of another double ratio

219
00:10:48.365 --> 00:10:51.105
or proportion that holds between four numbers.

220
00:10:52.015 --> 00:10:54.065
They're the number 6, 8, 9,

221
00:10:54.125 --> 00:10:57.505
and 12 that were actually called in Plato's time.

222
00:10:57.845 --> 00:11:01.745
The Harmonia was also called the Musical TEUs.

223
00:11:02.965 --> 00:11:05.705
Now, this structure was discussed in Plato's time

224
00:11:06.645 --> 00:11:10.265
and the series, this series of numbers coincided

225
00:11:10.265 --> 00:11:12.985
with the divisions of a, a vibrating string

226
00:11:13.375 --> 00:11:17.065
that would emit certain consonant notes,

227
00:11:17.065 --> 00:11:21.405
that now the four numbers represented the notes

228
00:11:21.405 --> 00:11:23.925
of the a tonic note of a scale followed

229
00:11:24.105 --> 00:11:27.045
by what's called the perfect fourth, as in say,

230
00:11:27.285 --> 00:11:28.725
C two F in our modern scale.

231
00:11:29.235 --> 00:11:31.165
Then a perfect fifth C two G,

232
00:11:31.385 --> 00:11:35.125
and then an octave up to the next C Right now, the fourth

233
00:11:35.185 --> 00:11:37.525
and fifth of this series correspond

234
00:11:37.525 --> 00:11:38.725
to the so-called the Harmonic

235
00:11:38.725 --> 00:11:41.765
and Arithmetic means of this interval,

236
00:11:42.425 --> 00:11:44.565
and the octaves were elated in a series,

237
00:11:44.565 --> 00:11:46.245
which is called a geometric series,

238
00:11:46.705 --> 00:11:49.445
and the means of that series are called geometric means.

239
00:11:50.265 --> 00:11:52.965
Now, these all come from that original period

240
00:11:53.085 --> 00:11:55.405
of Pythagorean, number theory

241
00:11:55.625 --> 00:11:58.165
and music theory,

242
00:11:58.305 --> 00:12:02.005
and a number of, later neo platonists

243
00:12:03.255 --> 00:12:07.275
had identified this, so-called Harmonia, this structure

244
00:12:07.295 --> 00:12:10.955
of four numbers as being the most beautiful bond

245
00:12:11.785 --> 00:12:15.875
that Plato held to be holding the sort

246
00:12:15.875 --> 00:12:18.955
of universe together in this logistics structure.

247
00:12:20.495 --> 00:12:25.275
Now, these three means are by no means sort of, sort

248
00:12:25.275 --> 00:12:28.115
of just just related to Pythagorean music theory.

249
00:12:28.115 --> 00:12:32.595
They're still used in statistics in modern mathematics,

250
00:12:32.595 --> 00:12:35.275
and they have applications in a range of, areas

251
00:12:35.335 --> 00:12:37.515
of mathematics beyond music theory.

252
00:12:38.255 --> 00:12:41.635
Um, and I became interested in their role in projective

253
00:12:41.635 --> 00:12:44.915
geometry to make this sort of link to modern mathematics

254
00:12:45.465 --> 00:12:48.835
Godfrey Live, and it's is a central figure in my account.

255
00:12:49.015 --> 00:12:51.995
Um, he's the sort of linchpin, really, between the ancient

256
00:12:51.995 --> 00:12:53.115
and the modern stuff.

257
00:12:54.055 --> 00:12:56.755
Now, liveness was a, a great innovator in logic,

258
00:12:58.025 --> 00:12:59.965
and it's, it's often thought.

259
00:13:00.785 --> 00:13:04.085
Um, he's often thought of as the originator of modern

260
00:13:04.845 --> 00:13:06.885
mathematical, so-called as opposed

261
00:13:06.885 --> 00:13:08.805
to traditional Aristotelian logic,

262
00:13:10.505 --> 00:13:13.005
but Leiter's logical work was not particularly

263
00:13:13.005 --> 00:13:14.405
appreciated in his time.

264
00:13:14.785 --> 00:13:17.725
Um, he, his logical writings weren't widely known.

265
00:13:17.755 --> 00:13:19.845
They were known by small groups of people,

266
00:13:19.845 --> 00:13:21.485
usually mathematicians more than,

267
00:13:21.595 --> 00:13:23.085
certainly more than philosophers.

268
00:13:24.605 --> 00:13:27.905
At the end of the 19th century, he became recognized,

269
00:13:27.925 --> 00:13:30.025
and it was, it was recognized

270
00:13:30.025 --> 00:13:34.105
that he had in anticipated many of the features of the type

271
00:13:34.105 --> 00:13:37.825
of logic that actually developed in the mid 19th century,

272
00:13:38.925 --> 00:13:41.905
especially by the character George Bull.

273
00:13:42.085 --> 00:13:44.305
Now, of course, everyone knows of bull these days

274
00:13:44.305 --> 00:13:47.585
because of his importance for computer com.

275
00:13:47.585 --> 00:13:50.265
You know, the theory of computers, it's the type of logic

276
00:13:50.295 --> 00:13:53.465
that is applied in, computer theory

277
00:13:54.015 --> 00:13:55.985
rule develops this type of,

278
00:13:56.155 --> 00:13:59.665
logic in the 19th century in which he applies the non-Greek

279
00:13:59.665 --> 00:14:00.945
discipline of algebra.

280
00:14:01.495 --> 00:14:03.305
That algebra comes from Indian

281
00:14:03.385 --> 00:14:05.705
and Arab roots, not Greek roots.

282
00:14:06.005 --> 00:14:08.785
He applies it to the Aristotelian Syllogism.

283
00:14:09.525 --> 00:14:11.305
Um, and it's recognized now

284
00:14:11.305 --> 00:14:15.225
that Livens anticipated this sort of move,

285
00:14:15.405 --> 00:14:16.785
and it looks like liven.

286
00:14:16.785 --> 00:14:19.585
It's modeled this application of algebra to

287
00:14:20.485 --> 00:14:22.465
the syllogism on the model

288
00:14:23.005 --> 00:14:27.265
in which Descartes had earlier applied algebra to geometry

289
00:14:28.005 --> 00:14:32.065
to establish what we now know as, analytic geometry,

290
00:14:32.085 --> 00:14:33.865
the sort of geometry you do with.

291
00:14:33.925 --> 00:14:35.865
So Cartesian coordinates, right?

292
00:14:36.065 --> 00:14:40.585
X and y coordinates to give values to, to give effectively,

293
00:14:40.645 --> 00:14:44.505
to give equations to, to figures, geometric figures,

294
00:14:45.245 --> 00:14:46.585
modern type of geometry.

295
00:14:46.935 --> 00:14:49.745
When Hagels account of the Sistic logic in Book three,

296
00:14:49.885 --> 00:14:51.710
the Science of Logic is discussed,

297
00:14:52.185 --> 00:14:53.725
it standardly just assumed

298
00:14:53.725 --> 00:14:57.365
that he's talking about Aristo Aristotle syllogism,

299
00:14:57.505 --> 00:14:59.605
and that he's a mine,

300
00:14:59.605 --> 00:15:04.085
perhaps a minor variant on Aristo Aristotle's Sistic logic.

301
00:15:04.425 --> 00:15:06.045
And this seemed to me to be wrong,

302
00:15:06.465 --> 00:15:09.805
and it ignored the degree to which he had been influenced

303
00:15:09.945 --> 00:15:12.685
by, LNI,

304
00:15:14.145 --> 00:15:18.565
and that Hagel had incorporated something like L NI's logic

305
00:15:18.915 --> 00:15:19.925
into his own way

306
00:15:20.145 --> 00:15:22.325
and his own ways of thinking about syllogisms,

307
00:15:22.535 --> 00:15:26.685
which reflected elements actually of Plato's syllogisms.

308
00:15:27.665 --> 00:15:30.405
Now, LNI logic may not have been widely understood

309
00:15:30.465 --> 00:15:32.085
by German philosophers at the time,

310
00:15:32.425 --> 00:15:34.725
but it was effectively the basis of the type

311
00:15:34.725 --> 00:15:36.725
of logic taught at the Tubing

312
00:15:36.725 --> 00:15:39.085
and Seminary when Hagel was a student.

313
00:15:39.085 --> 00:15:43.525
There, there was a character named Got Freed Pke,

314
00:15:43.525 --> 00:15:45.565
and he introduced important modifications

315
00:15:45.565 --> 00:15:49.125
to Aristotle's Logic that Hagel himself adopted

316
00:15:51.155 --> 00:15:52.935
Pke is now sometimes discussed in the

317
00:15:52.935 --> 00:15:54.015
history of computation.

318
00:15:54.675 --> 00:15:57.735
Um, he's, and he's regarded as quite a figure in this,

319
00:15:57.955 --> 00:16:02.615
in this tradition Or one of one

320
00:16:02.615 --> 00:16:05.855
of these differences to Aristotelian logic that,

321
00:16:07.195 --> 00:16:10.695
PKE introduced was effectively to introduce a distinction

322
00:16:11.525 --> 00:16:15.055
that has appeared in modern logic in 20th century logic

323
00:16:15.755 --> 00:16:19.135
as a semantic distinction between the role of proper names

324
00:16:19.835 --> 00:16:21.095
and definite descriptions.

325
00:16:21.285 --> 00:16:25.295
This is something bound up with sort of arguments about

326
00:16:25.295 --> 00:16:26.535
between Bur Burton Russell

327
00:16:26.635 --> 00:16:29.695
and later on modal Logicians like Saul Kripke

328
00:16:29.695 --> 00:16:31.335
and people like that in the 20th century.

329
00:16:32.075 --> 00:16:36.935
But that, but, but that def distinction goes back to,

330
00:16:37.525 --> 00:16:40.175
certainly goes back to plu K He uses it

331
00:16:40.175 --> 00:16:43.255
and Hagel adopts, adopts it from Pke.

332
00:16:43.795 --> 00:16:46.615
And this distinction, I think, lines up in many ways

333
00:16:46.885 --> 00:16:48.215
with the way Hagel employed.

334
00:16:48.235 --> 00:16:51.575
The idea of the three musical memes in the syllogism

335
00:16:51.875 --> 00:16:56.495
and the way that he used Plato against, Aristotle.

336
00:16:57.205 --> 00:16:59.855
Aristotle doesn't make distinctions it's bet like that

337
00:16:59.855 --> 00:17:03.215
between proper names and general descriptions, right?

338
00:17:03.335 --> 00:17:05.295
I mean, it doesn't really feature in his logic.

339
00:17:07.325 --> 00:17:10.305
Now, lo Litz had an astonishing range of interests.

340
00:17:10.425 --> 00:17:12.855
I mean, and they're all over the place

341
00:17:12.915 --> 00:17:14.695
and it's hard to link up his mind,

342
00:17:14.725 --> 00:17:17.575
changed his mind on different topics here and there,

343
00:17:17.595 --> 00:17:19.415
and he's got one logic for one year

344
00:17:19.475 --> 00:17:21.855
and a different logic for the next year and so on.

345
00:17:23.035 --> 00:17:25.935
Um, and one of his interests was in this new,

346
00:17:26.155 --> 00:17:28.295
or this, this new type of geometry,

347
00:17:28.365 --> 00:17:30.015
what we call projective geometry.

348
00:17:30.745 --> 00:17:32.975
Projective geometry had a sort of a brief

349
00:17:34.095 --> 00:17:35.455
introduction in the 17th century

350
00:17:36.195 --> 00:17:38.935
by a French mathematician named Ard Deza,

351
00:17:39.955 --> 00:17:42.055
and, one of de a's followers

352
00:17:42.055 --> 00:17:44.855
who was actually Blaze Pascal, who, you know,

353
00:17:44.855 --> 00:17:46.015
a lot of people know about.

354
00:17:47.785 --> 00:17:51.365
Um, and Deza had actually been influenced in his approach,

355
00:17:51.585 --> 00:17:55.605
in his approach by the optics of Hegel's hero Kepper, sorry,

356
00:17:55.705 --> 00:17:58.125
Kepper as an important sort of person

357
00:17:58.345 --> 00:18:02.285
to feed into this tradition of, projective geometry.

358
00:18:03.315 --> 00:18:05.445
Both Dega, both de Arg

359
00:18:05.825 --> 00:18:09.165
and Leni had been interested in the relation of the study

360
00:18:09.225 --> 00:18:11.365
of perspective,

361
00:18:12.675 --> 00:18:15.965
that had emerged out of, theories

362
00:18:15.965 --> 00:18:18.365
of artistic representation in the Renaissance,

363
00:18:18.825 --> 00:18:21.325
and they tried to relate that to the geometry

364
00:18:21.325 --> 00:18:22.725
of three dimensional space.

365
00:18:23.105 --> 00:18:27.405
And somehow that was seen as relating to this, new type

366
00:18:27.405 --> 00:18:29.645
of geometry, projective geometry.

367
00:18:30.885 --> 00:18:35.205
Lyme had explored a variety of geometrical ideas relevant

368
00:18:35.205 --> 00:18:36.525
to the understanding of space.

369
00:18:37.185 --> 00:18:39.685
The one that I take up in the book is approach is a,

370
00:18:39.685 --> 00:18:43.245
is an approach that he called analysis CTUs,

371
00:18:43.425 --> 00:18:45.645
or the analysis of situations.

372
00:18:46.785 --> 00:18:51.165
And his analysis CTUs was meant as a type of alternative

373
00:18:51.185 --> 00:18:53.205
to dakar's analytic geometry.

374
00:18:53.505 --> 00:18:56.285
It was supposed to be a non metrical alternative

375
00:18:56.585 --> 00:18:58.405
to dakar's analytic geometry.

376
00:18:58.555 --> 00:19:01.605
Descartes sort of re again, re reduces geometry

377
00:19:01.825 --> 00:19:04.325
to arithmetic via algebra.

378
00:19:05.705 --> 00:19:07.325
Um, but they,

379
00:19:07.465 --> 00:19:11.685
but, Leni analysis CTUs has features similar

380
00:19:11.785 --> 00:19:16.205
to projective geometry of Deza and his fellow Blaze Pascal.

381
00:19:16.985 --> 00:19:18.365
In fact, I mean, interestingly,

382
00:19:19.035 --> 00:19:23.085
Leni is the only person really with the type of capacity

383
00:19:23.085 --> 00:19:24.365
to understand what was going on,

384
00:19:24.745 --> 00:19:29.365
who read this important document by Pascal called,

385
00:19:29.365 --> 00:19:32.805
it's about his mystical hexagram,

386
00:19:32.915 --> 00:19:34.325
that was subsequently lost.

387
00:19:34.825 --> 00:19:37.845
And all we know about it is from a couple of letters

388
00:19:37.915 --> 00:19:40.245
that Decart wrote to people about, about

389
00:19:40.245 --> 00:19:41.805
what was in this document.

390
00:19:41.945 --> 00:19:45.725
And, so he was clearly influenced by this tradition of,

391
00:19:45.745 --> 00:19:46.925
projective geometry.

392
00:19:47.905 --> 00:19:49.845
Others have traced the influence

393
00:19:49.865 --> 00:19:52.885
of analysis set on modern branches of geometry such

394
00:19:52.885 --> 00:19:53.965
as topology.

395
00:19:54.705 --> 00:19:57.685
Um, Leni logic is certainly criticized

396
00:19:57.685 --> 00:19:59.485
by hagel in the science of logic,

397
00:19:59.785 --> 00:20:03.165
but I think what is criticized is the idea of the type

398
00:20:03.165 --> 00:20:07.205
of reductive version of it that we find that's aligned

399
00:20:07.205 --> 00:20:08.525
to dakar's geometry

400
00:20:09.345 --> 00:20:12.605
and to which this alternative, idea

401
00:20:12.605 --> 00:20:15.805
of analysis CTUs would be seen as offering an alternative.

402
00:20:16.625 --> 00:20:20.195
Because it's a fairly big departure from the way these

403
00:20:20.195 --> 00:20:23.155
things have been discussed over the last 20 or 30 years.

404
00:20:23.215 --> 00:20:25.875
The time that I've been involved in Hagel scholarship,

405
00:20:26.295 --> 00:20:29.075
I'm trying to establish at least a primer Facey case

406
00:20:29.655 --> 00:20:32.035
for this type of platonic Pythagorean

407
00:20:32.465 --> 00:20:33.955
dimension to Hale's logic.

408
00:20:35.605 --> 00:20:39.025
In recent times, Hagel interpreters have tended to focus

409
00:20:39.485 --> 00:20:43.425
around, there's been a debate around the question

410
00:20:43.425 --> 00:20:46.665
of whether Hegel's logic should be seen as continuous with

411
00:20:46.665 --> 00:20:49.845
that developed by Emmanuel KT before Hagel,

412
00:20:50.425 --> 00:20:53.645
or whether it should be seen as that, like the, logic

413
00:20:53.665 --> 00:20:55.525
of Aristotle, his logistic logic.

414
00:20:56.425 --> 00:20:59.605
And I think this restriction has led to diminishing returns.

415
00:21:01.025 --> 00:21:05.965
Um, there have been people who have suggested,

416
00:21:06.145 --> 00:21:10.965
the role, the important role of Plato in Hegel's logic,

417
00:21:11.105 --> 00:21:13.285
but they haven't been given much

418
00:21:13.285 --> 00:21:15.325
of a voice in this recent debate.

419
00:21:16.075 --> 00:21:18.965
Putting Hegel's logic in this relatively new perspective,

420
00:21:19.245 --> 00:21:21.125
I think of the sort of platonic

421
00:21:22.125 --> 00:21:24.605
Pythagorean slash musical perspective,

422
00:21:25.285 --> 00:21:27.365
I hope will shed light on other features

423
00:21:27.365 --> 00:21:30.605
of Hegel's philosophy, given the fact that he thinks of

424
00:21:31.135 --> 00:21:33.885
logic itself as the foundation of all the rest

425
00:21:33.885 --> 00:21:36.005
of his philosophies, his philosophy of nature,

426
00:21:36.465 --> 00:21:39.645
his philosophy of sort of human subjectivity or spirit

427
00:21:40.225 --> 00:21:43.685
or human or social and cultural existence and so on.

428
00:21:45.385 --> 00:21:48.725
But I suppose mostly I've tried to aim at establishing,

429
00:21:49.095 --> 00:21:52.765
again, at a prime, a prima facie case for the relevance

430
00:21:52.765 --> 00:21:56.365
of Hegel's logic as it's currently practiced as,

431
00:21:56.465 --> 00:21:58.525
as in modern movements of logic.

432
00:21:58.795 --> 00:22:01.045
Because even Hagels don't seem

433
00:22:01.085 --> 00:22:02.525
to be particularly interested in that.

434
00:22:02.595 --> 00:22:04.325
They, they're quite happy to think

435
00:22:04.325 --> 00:22:07.405
that Hagel doesn't have any bearing on modern logic,

436
00:22:07.435 --> 00:22:10.005
whereas a lot of modern logic logicians just

437
00:22:10.005 --> 00:22:11.645
dismiss Hagel outright.

438
00:22:12.065 --> 00:22:15.765
So I'm trying to establish the, I'm trying to establish,

439
00:22:16.105 --> 00:22:19.885
a place for Hagel in the history of logic

440
00:22:20.145 --> 00:22:21.245
as it's practiced now.

441
00:22:21.665 --> 00:22:24.245
Um, which I think it's, I think he deserves

442
00:22:24.545 --> 00:22:28.285
to be much more, acknowledged as a sort of a,

443
00:22:28.525 --> 00:22:30.765
a player in this, arena.

444
00:22:30.985 --> 00:22:34.805
He hasn't been, he hasn't been acknowledged as such up

445
00:22:34.805 --> 00:22:36.445
to the present, I hope

446
00:22:36.445 --> 00:22:39.405
that readers will just find the material interesting

447
00:22:39.505 --> 00:22:42.405
and as interesting as I found it when I sort

448
00:22:42.405 --> 00:22:45.845
of stumbled across it, I mean, it's just a fascinating sort

449
00:22:45.845 --> 00:22:47.245
of, stuff.

450
00:22:47.485 --> 00:22:51.445
I mean, everyone who gets seduced by projective geometry

451
00:22:51.985 --> 00:22:56.365
and, everyone I've ever spoken to who's got a taste

452
00:22:56.465 --> 00:22:59.045
for it, finds it really hard to let go.

453
00:22:59.115 --> 00:23:03.165
It's a type of geometry that's been, sadly sort

454
00:23:03.165 --> 00:23:04.565
of neglected in the 20th century,

455
00:23:04.625 --> 00:23:07.325
but it had a huge role in the 19th century,

456
00:23:07.785 --> 00:23:10.285
and it fed into all sorts of,

457
00:23:10.675 --> 00:23:13.965
important discoveries at the end of the 19th century.

458
00:23:14.145 --> 00:23:17.405
It fed into Einstein the type of geometries

459
00:23:17.405 --> 00:23:20.245
that went into Einstein's physics and so forth.

460
00:23:20.265 --> 00:23:23.445
So it has a, a huge relevance for modern thought,

461
00:23:23.555 --> 00:23:27.245
even though as a particular type of, geometry,

462
00:23:27.275 --> 00:23:29.525
it's been sort of abandoned a bit.

463
00:23:30.125 --> 00:23:35.085
I tend to hope that, putting hagel in this context well,

464
00:23:35.625 --> 00:23:38.525
people will start to see hagel in a different light, right?

465
00:23:38.525 --> 00:23:43.285
They just won't see him as a, this one-off weird, person

466
00:23:43.425 --> 00:23:45.765
who was sort of trying to invent stuff out of nowhere.

467
00:23:46.685 --> 00:23:49.885
I hope they'll see him as a part of a tradition of logic

468
00:23:50.315 --> 00:23:52.005
that even draws on mathematics.

469
00:23:52.145 --> 00:23:56.365
Um, hegelians have been very Charlie of thinking of hagel

470
00:23:56.425 --> 00:23:58.165
as having anything to do with mathematics,

471
00:23:58.255 --> 00:24:00.405
which I think is very misleading.

472
00:24:02.435 --> 00:24:04.975
Um, at the same time I've discovered that there's something

473
00:24:04.975 --> 00:24:07.615
of an interest in Hegel's logic among certain groups

474
00:24:07.635 --> 00:24:10.735
of mathematicians, especially those

475
00:24:11.355 --> 00:24:14.655
in those very philosophical areas of mathematics that are

476
00:24:15.475 --> 00:24:19.655
far too sort of complex for me to get my get a sense of,

477
00:24:20.355 --> 00:24:22.895
such as contemporary category theory.

478
00:24:23.635 --> 00:24:25.375
Um, category theory is a theory

479
00:24:25.375 --> 00:24:28.575
that arose in the 20th century from around the 1940s,

480
00:24:29.555 --> 00:24:32.815
and it's become a type of rival to set theory

481
00:24:33.635 --> 00:24:37.055
as if you like, the most general language within mathematics

482
00:24:37.165 --> 00:24:39.655
with within which mathematics is explored.

483
00:24:40.595 --> 00:24:43.535
Now, I became aware that there were certain sort

484
00:24:43.535 --> 00:24:45.855
of fairly important category theorists

485
00:24:46.275 --> 00:24:49.455
who were saying pretty interesting things about hagel,

486
00:24:49.595 --> 00:24:52.295
and that's one of the things that got me interested in it.

487
00:24:52.915 --> 00:24:56.015
Um, I realized I would never go to learn enough mathematics

488
00:24:56.435 --> 00:25:00.255
to say anything interesting about hagel and category theory,

489
00:25:00.315 --> 00:25:04.255
but I thought I perhaps could say something about hagel

490
00:25:04.275 --> 00:25:07.575
and the type of mathematics that category theory is said

491
00:25:07.575 --> 00:25:10.735
to have its origins in which are these same 19th century

492
00:25:10.825 --> 00:25:13.135
areas such as, projective geometry

493
00:25:13.395 --> 00:25:15.175
and so on that we've been discussing.

494
00:25:16.195 --> 00:25:19.535
So imploring, exploring some of these issues from

495
00:25:20.085 --> 00:25:23.055
hegel's from the angle of hegel's thinking I thought

496
00:25:23.665 --> 00:25:26.775
might be relevant to those who'd been looking at these same

497
00:25:26.775 --> 00:25:29.255
things about the history of mathematics,

498
00:25:29.565 --> 00:25:31.495
from the more mathematical end of things

499
00:25:31.595 --> 00:25:34.895
and from the end of, the perspective of category theory.

Greek Foundations of Hegel's Logic

00:00:00 - So I was primarily interested in, the Greek influences on Hegel's logic. I've been working on Hagel for too many years now, and I became interested in what is now known about the origins of Aristotle's Sistic logic, and in particular its relation to mathematics As it was practiced in that period, I was motivated by a claim that Hagel makes in his lectures on the history of philosophy, where he suggests that Aristotle's logic had been derived from the account of the universe that,
00:00:34 - You know, the idea was that Aristotle derives his account of the logistic structure of inferences and so forth from this account and that, but that Aristo had simplified Plato's account, and Hagel says quite mysteriously he had reduced Plato's two means to one. so the question is, well, what are these means that he's talking about? I think it became, pretty obvious that what,
00:01:04 - or middle terms of these proportions that found in Pythagorean Music theory. and that is actually structuring Plato's account of the universe that's given in the time maus. Luckily for me, there's been a lot of work done over the last 50 years or so on the history of Greek mathematics, and in particular a lot of really good work on Pythagorean music theory. So I set out to examine the, the intelligibility
00:01:34 - of hagel sketches right against what we now know. So many people have comment commented over the years that the influence, on the influence of, Greek geometry on Aristotle's logic, and that seems pretty plausible. We know a lot of the geometric theorems that ended up in Euclid's, elements, the great work that came from, you know, after Plato and Aristotle's time, we know that a lot of those had derived from the mathematics being practiced in
00:02:05 - of the fourth century when Plato established his academy. And of course, the idea of proof from axioms and definitions to theories is central to Euclidean geometry. So Aristotle's logic concerns syllogisms in which pairs of premises, like two statements lead to by necessity, a conclusion. And it seems natural that he would've in some way been drawn to this, existing idea of the proof structures in geometry.
00:02:38 - that Aristotle had actually employed diagrams or did teach logic, in a similar way to the way in which diagrams are employed in geometric proofs. So all that seemed pretty uncontroversial, but it was what what takes us beyond that to me, which was interesting. Plato had founded his academy, which is sort of like the first university, if you like, around 3 87, you know, before the common era BC

Influences on Aristotle's Development of Geometry

00:03:07 - to have joined at about 20 years later as a teenager. So he was considerably, you know, Plato's Junior Aristotle has been particularly described as having been influenced by a mathematician named Oxis of Tus. A Oxis was a little older than Aristotle, but he apparently joined the Academy around the same time, and he's one of the really, sort of big contributors to Euclidean geometry in particular. Oxis is thought to have introduced
00:03:44 - the contents of this particular book, book five, which is to do with proportions and ratios, and this is seen to be the sort of major move which established geometry as a type of autonomous discipline, autonomous from arithmetic, which the Aris, which the, which the Greeks weren't very good at. The Greek arithmetic more or less came from, Mesopotamia, and they never really developed it, so they had to break away from arithmetic to develop geometry in the way that they did.
00:04:16 - of geometry and arithmetic, and that it was discovered that, ratios of line lengths such the sort of things that geometry is interested in, right? Like the ratio between the side and the diagonal of a square could actually couldn't be represented by ratios of numbers. in modern terms, would would say that the ratio re required irrational numbers, right square root of two,
00:04:45 - of number, so there was that problem. So a different approach was necessary, and Oxis is seen as the key person there. Now, Plato would've been in his early sixties by the time of these sort of discoveries, and his mathematics seems to have been much more dependent on the approach of earlier Pythagorean. Now, people such as, there's a character named Phil La of Croton Russ, roughly a contemporary of Socrates and age, and a person called Tis of Toum, who was roughly Plato's,
00:05:23 - They were the major music theorists of the time. This Pythagorean tradition with its musical references seems to be largely eclipsed by the geometry of Oxis by the time that developed, however, it persisted more or less in the background of Euclidean geometry and in other aspects of geometry at the time, and it became linked to later geometric discoveries.
00:05:52 - so-called Connic sections. If you, if you think they're figures like circles, ellipses, parabolas and hyperbola, they're all linked up. The Greeks used to think of them as being, you'd get them by sectioning a cone in on different angles. So the Pythagorean tradition didn't entirely disappear, and in fact, it seems to have been revived in certain developments in modern mathematics that I think Hagel was, interested in,

Pythagorean Influence on Hegel's Logic

00:06:21 - Mayas the fictional character time Mayas who's it's thought that he may be, based on Akes or, or, you know, someone else around that time to Mayas describes the way the cosmos was put together by the great artifice, you know, sort of Plato's God as being held together by what Plato described as the most beautiful bond. Now that the structuring of ratios and proportions in this account was taken from Pythagorean. Music theory, as I mentioned,
00:06:53 - of the Spheres tradition. Now, in a seemingly sensible, people always dismiss this. Aristotle certainly dismissed it, the idea that the planets give up heavenly sort of notes that are all harmonious, right? Aristotle dismissed it as a beer metaphor. This tradition persisted up to the Renaissance, and in fact, later it was taken seriously by the great German astronomer, Johannes Kepper. Kepper was actually from Hagels hometown,
00:07:26 - And Hagel thought that Kepler was clearly onto something with this, sort of platonic Pythagorean approach to cosmology. And Hagel was roundly criticized for buying into this tradition. But I think that that hagel took this in a more logical than a more cosmological sense, right? He was thinking of the structuring of Kepler's thinking it had something to do with the way that logic, the,
00:07:55 - is captured in our scientific explanations. So his support of Plato's logic with its musical means over Aristotle's more reduced Euclidean form was meant to be, more adequate to this type of, modern astronomy. I think what struck me about Hegel's claim about Aristotle's logic as deriving from Plato's musical type of logic, is the fact that about a hundred years ago in the 1930s,
00:08:32 - that the vocabulary that Aristotle introduces for his logic, the specialist vocabulary, had actually been taken from the vocabulary of Pythagoria and music theory. Now, later, another historian of Greek mathematics, a a Russian named Abo also looked at the way that Pythagorean music had been brought into the geometric conceptions of ratios, the sort of important stuff
00:09:01 - these were the sorts of links that har that, that Hagel had been exploring in some sort of way between the lines of his account. So Hagel, I came to believe thought that a logic like Aristotle's model predominantly on Euclidean geometry needed to be supplemented by context concepts that drew on this non-Euclidean tradition with its earlier pla, platonic and Pythagorean roots.

Connections Between Geometry and Greek Musical Theory

00:09:31 - that certain modern trains of thought had also drawn upon this tradition. in the book, I argue that I think that he was right. The mathematical approach to geometry that's now known as projective geometry, which was a huge development in the 19th century, actually had conceptual links with Greek musical theory, via its links to the theory of comic sections.
00:10:02 - of the central proportions of projective geometry became called, became known, known as the harmonic cross ratio. and it goes back to what the Greek astronomer apollonius working after the time of Euclid and drawing on some of these musical ideas. in, in his work on the Connic sections had described as the harmonic division of a line, a harmonic division of a line divided it into four points
00:10:34 - and assigned numbers were assigned to those points that could be arranged in a type of double ratio. Now, this double ratio, as I argue in the book, is in fact a generalization of another double ratio or proportion that holds between four numbers. They're the number 6, 8, 9, and 12 that were actually called in Plato's time. The Harmonia was also called the Musical TEUs.
00:11:06 - and the series, this series of numbers coincided with the divisions of a, a vibrating string that would emit certain consonant notes, that now the four numbers represented the notes of the a tonic note of a scale followed by what's called the perfect fourth, as in say, C two F in our modern scale. Then a perfect fifth C two G, and then an octave up to the next C Right now, the fourth
00:11:37 - to the so-called the Harmonic and Arithmetic means of this interval, and the octaves were elated in a series, which is called a geometric series, and the means of that series are called geometric means. Now, these all come from that original period of Pythagorean, number theory and music theory, and a number of, later neo platonists had identified this, so-called Harmonia, this structure
00:12:11 - that Plato held to be holding the sort of universe together in this logistics structure. Now, these three means are by no means sort of, sort of just just related to Pythagorean music theory. They're still used in statistics in modern mathematics, and they have applications in a range of, areas of mathematics beyond music theory.

Godfrey Live: Bridging Ancient and Modern Logic

00:12:41 - geometry to make this sort of link to modern mathematics Godfrey Live, and it's is a central figure in my account. he's the sort of linchpin, really, between the ancient and the modern stuff. Now, liveness was a, a great innovator in logic, and it's, it's often thought. he's often thought of as the originator of modern mathematical, so-called as opposed to traditional Aristotelian logic,
00:13:13 - appreciated in his time. he, his logical writings weren't widely known. They were known by small groups of people, usually mathematicians more than, certainly more than philosophers. At the end of the 19th century, he became recognized, and it was, it was recognized that he had in anticipated many of the features of the type of logic that actually developed in the mid 19th century, especially by the character George Bull.
00:13:44 - because of his importance for computer com. You know, the theory of computers, it's the type of logic that is applied in, computer theory rule develops this type of, logic in the 19th century in which he applies the non-Greek discipline of algebra. That algebra comes from Indian and Arab roots, not Greek roots. He applies it to the Aristotelian Syllogism. and it's recognized now that Livens anticipated this sort of move,
00:14:16 - It's modeled this application of algebra to the syllogism on the model in which Descartes had earlier applied algebra to geometry to establish what we now know as, analytic geometry, the sort of geometry you do with. So Cartesian coordinates, right? X and y coordinates to give values to, to give effectively, to give equations to, to figures, geometric figures, modern type of geometry.
00:14:49 - the Science of Logic is discussed, it standardly just assumed that he's talking about Aristo Aristotle syllogism, and that he's a mine, perhaps a minor variant on Aristo Aristotle's Sistic logic. And this seemed to me to be wrong, and it ignored the degree to which he had been influenced by, LNI, and that Hagel had incorporated something like L NI's logic
00:15:20 - and his own ways of thinking about syllogisms, which reflected elements actually of Plato's syllogisms. Now, LNI logic may not have been widely understood by German philosophers at the time, but it was effectively the basis of the type of logic taught at the Tubing and Seminary when Hagel was a student. There, there was a character named Got Freed Pke, and he introduced important modifications to Aristotle's Logic that Hagel himself adopted

Evolution of Logic and Geometry in Philosophy

00:15:52 - history of computation. he's, and he's regarded as quite a figure in this, in this tradition Or one of one of these differences to Aristotelian logic that, PKE introduced was effectively to introduce a distinction that has appeared in modern logic in 20th century logic as a semantic distinction between the role of proper names and definite descriptions.
00:16:25 - between Bur Burton Russell and later on modal Logicians like Saul Kripke and people like that in the 20th century. But that, but, but that def distinction goes back to, certainly goes back to plu K He uses it and Hagel adopts, adopts it from Pke. And this distinction, I think, lines up in many ways with the way Hagel employed. The idea of the three musical memes in the syllogism
00:16:57 - Aristotle doesn't make distinctions it's bet like that between proper names and general descriptions, right? I mean, it doesn't really feature in his logic. Now, lo Litz had an astonishing range of interests. I mean, and they're all over the place and it's hard to link up his mind, changed his mind on different topics here and there, and he's got one logic for one year and a different logic for the next year and so on.
00:17:26 - or this, this new type of geometry, what we call projective geometry. Projective geometry had a sort of a brief introduction in the 17th century by a French mathematician named Ard Deza, and, one of de a's followers who was actually Blaze Pascal, who, you know, a lot of people know about. and Deza had actually been influenced in his approach, in his approach by the optics of Hegel's hero Kepper, sorry,
00:17:58 - to feed into this tradition of, projective geometry. Both Dega, both de Arg and Leni had been interested in the relation of the study of perspective, that had emerged out of, theories of artistic representation in the Renaissance, and they tried to relate that to the geometry of three dimensional space. And somehow that was seen as relating to this, new type
00:18:30 - Lyme had explored a variety of geometrical ideas relevant to the understanding of space. The one that I take up in the book is approach is a, is an approach that he called analysis CTUs, or the analysis of situations. And his analysis CTUs was meant as a type of alternative to dakar's analytic geometry. It was supposed to be a non metrical alternative to dakar's analytic geometry.

Hegel's Logic and Its Mathematical Foundations

00:19:01 - to arithmetic via algebra. but they, but, Leni analysis CTUs has features similar to projective geometry of Deza and his fellow Blaze Pascal. In fact, I mean, interestingly, Leni is the only person really with the type of capacity to understand what was going on, who read this important document by Pascal called,
00:19:32 - that was subsequently lost. And all we know about it is from a couple of letters that Decart wrote to people about, about what was in this document. And, so he was clearly influenced by this tradition of, projective geometry. Others have traced the influence of analysis set on modern branches of geometry such as topology. Leni logic is certainly criticized by hagel in the science of logic,
00:20:03 - of reductive version of it that we find that's aligned to dakar's geometry and to which this alternative, idea of analysis CTUs would be seen as offering an alternative. Because it's a fairly big departure from the way these things have been discussed over the last 20 or 30 years. The time that I've been involved in Hagel scholarship, I'm trying to establish at least a primer Facey case for this type of platonic Pythagorean
00:20:35 - In recent times, Hagel interpreters have tended to focus around, there's been a debate around the question of whether Hegel's logic should be seen as continuous with that developed by Emmanuel KT before Hagel, or whether it should be seen as that, like the, logic of Aristotle, his logistic logic. And I think this restriction has led to diminishing returns. there have been people who have suggested,
00:21:11 - but they haven't been given much of a voice in this recent debate. Putting Hegel's logic in this relatively new perspective, I think of the sort of platonic Pythagorean slash musical perspective, I hope will shed light on other features of Hegel's philosophy, given the fact that he thinks of logic itself as the foundation of all the rest of his philosophies, his philosophy of nature,
00:21:40 - or human or social and cultural existence and so on. But I suppose mostly I've tried to aim at establishing, again, at a prime, a prima facie case for the relevance of Hegel's logic as it's currently practiced as, as in modern movements of logic. Because even Hagels don't seem to be particularly interested in that. They, they're quite happy to think that Hagel doesn't have any bearing on modern logic,

Hegel's Legacy in Logic and Geometry

00:22:10 - dismiss Hagel outright. So I'm trying to establish the, I'm trying to establish, a place for Hagel in the history of logic as it's practiced now. which I think it's, I think he deserves to be much more, acknowledged as a sort of a, a player in this, arena. He hasn't been, he hasn't been acknowledged as such up to the present, I hope that readers will just find the material interesting
00:22:42 - of stumbled across it, I mean, it's just a fascinating sort of, stuff. I mean, everyone who gets seduced by projective geometry and, everyone I've ever spoken to who's got a taste for it, finds it really hard to let go. It's a type of geometry that's been, sadly sort of neglected in the 20th century, but it had a huge role in the 19th century, and it fed into all sorts of,
00:23:14 - It fed into Einstein the type of geometries that went into Einstein's physics and so forth. So it has a, a huge relevance for modern thought, even though as a particular type of, geometry, it's been sort of abandoned a bit. I tend to hope that, putting hagel in this context well, people will start to see hagel in a different light, right? They just won't see him as a, this one-off weird, person
00:23:46 - I hope they'll see him as a part of a tradition of logic that even draws on mathematics. hegelians have been very Charlie of thinking of hagel as having anything to do with mathematics, which I think is very misleading. at the same time I've discovered that there's something of an interest in Hegel's logic among certain groups of mathematicians, especially those in those very philosophical areas of mathematics that are
00:24:20 - such as contemporary category theory. category theory is a theory that arose in the 20th century from around the 1940s, and it's become a type of rival to set theory as if you like, the most general language within mathematics with within which mathematics is explored. Now, I became aware that there were certain sort of fairly important category theorists
00:24:49 - and that's one of the things that got me interested in it. I realized I would never go to learn enough mathematics to say anything interesting about hagel and category theory, but I thought I perhaps could say something about hagel and the type of mathematics that category theory is said to have its origins in which are these same 19th century areas such as, projective geometry and so on that we've been discussing. So imploring, exploring some of these issues from

Insights on Mathematics and Category Theory

00:25:31 - might be relevant to those who'd been looking at these same things about the history of mathematics, from the more mathematical end of things and from the end of, the perspective of category theory.

Faculti AI can make mistakes

Research

Policy

Analysis