Slick cookie warning with a timer bar that just runs down. Genius. Transparent overlays. Proper line height. Who came up with this? This website is a pleasure.
Can anyone recommend a book that describes some potential destructive consequences of Cartesian thinking on everyday life in Western civilization (not just in the field of philosophy)? It seems possible to me that choosing the individual "subject" (the mind in mind-body dualism) as the starting point for Enlightenment discourse could have contributed in some way to the modern selfish solipsism we see in society today.
There's quite a strong criticism of how our principle of freedom of speech (as opposed to 'freedom of action') is built upon a Cartesian mind-body dualism written by Susan Brison (no book published, as far as I'm aware) building off the American legal and philosophical theory of Fred Schauer. I don't have links at hand at the moment, but it was an interesting read, as she argues that this 'Cartesian mistake' points out a glaring hole in the justification for a special constitutional amendment to freedom of speech.
Edit: I found a draft from 2018, but the author asks for it not to be cited. It was published earlier this year, as: "Free Speech Skepticism". Susan J. Brison. Kennedy Institute of Ethics Journal, Volume 31, Number 2, June 2021
Very interesting. I hadn’t actually thought about the _legalistic_ consequences of Cartesian dualism, but it’s a fascinating thought. Applied to free speech that’s a very unique problem - I’ll have to take a look. Thanks.
> Most readers will find Gadberry’s readings dazzlingly innovative; other less attentive and open-minded readers may view them as outlandish.
This kind of rhetorical gambit really bugs me. So if I find Gadberry unconvincing (or possibly even full of it) I'm close-minded and inattentive, am I?
It is shocking to most people to learn how absolutely recent spatializing numbers ("the number line") and time ("timelines") really is, and thus how deeply unnatural those now-familiar notions once seemed.
It is that process that made it possible for elementary schoolchildren to reasonably be expected to cope with algebra and world history, topics largely limited to the old world's most brilliant academics.
Decartes was in the thick of that transition.
Resistance to zero and negative numbers had to evaporate once the number line was understood.
In western mathematics? Maybe. In India analytic geometry was the norm and Indians used infinity and negative numbers for over a millennia before Europeans.
Yes. The West saw extremely limited contact with ancient Indian mathematics until quite recently. I think India itself largely forgot about its own mathematical tradition well before it was occupied by European powers.
The Wikipedia page on Analytic Geometry utterly fails to so much as mention classical Indian work on the topic. I presume some officious editor is keeping it well-scrubbed of any non-Western credits.
That’s pretty common. The issue is it’s hard to have well referenced scholarly sources for these things. Indians have a tendency to mystify their past. Such shame really. One of my long term goals in life is to create a authoritative, non religious, simple source for Indian Classics.
Hard to give a concrete source because it’s simply not there. See my comment in the sibling thread. I can make a case and you can judge for yourselves.
The key insight in analytic geometry is that geometric shapes and algebra are facets of the same underlying concept. This allowed Descartes to develop coordinate system. This in turn allowed development of calculus and everything else. This was a striking discovery in the western world.
The practice of using equations for geometry however was very common in India. Brahmagupta was the first to have a formula for area of cyclic quadrilateral in 6th century [2]. Notably the proof used trigonometry. Earlier similar result in West is Heron’s formula for the area of triangle. However the proof doesn’t use any algebra.
Bhaskara had devised ideas of differential calculus by 11th century [2]. A concrete example of this is the use of sine and cosine series in Kerala School of Mathematics [3].
This is really just the tip of the iceberg because majority of Indian mathematics is simply lost. What remains is poorly understood. But when reliable translations are available you can see that idea of using geometry and algebra interchangeably was common.
Maybe I’m biased. Curious to know what others make of this
> Brahmagupta was the first to have a formula for area of cyclic quadrilateral in 6th century
That's not analytic geometry.
> Bhaskara had devised ideas of differential calculus by 11th century [2]. A concrete example of this is the use of sine and cosine series in Kerala School of Mathematics [3].
Yeah read the distance and angle section. Describing geometric objects with equations in coordinate system is analytic geometry.
Anyway I’m not going to seriously discuss with you because it’s clear from your post history that your replies are avalanches of pedantry. If you think it has a specific meaning, elaborate it and identify the point of divergence or express what your view is. I don’t have an obligation to convince you.
Have you studied math? I know Greeks used angles and distances lol. The point is how. Did they describe them algebraically or did they do the Euclidean thing where they “take a compass and bisect and you’ll see each part is equal to this other side”. Euclid/Greeks used pure geometry. They might have had all geometric concepts. But they didn’t manipulate algebraic equations to obtain geometric insights. And yes that is analytic geometry
No, the example you gave is not analytic geometry. Read the definition again. The explicit use of a coordinate system is essential. Algebra is not sufficient.
> Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates *or formulae*. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems.
> Only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to geometry from other approaches.
From wiki of analytic geometry [2]:
> In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This *contrasts* with synthetic geometry.
I know you’re using this to dismiss my case but the first line of Wikipedia hardly a standard for definition. Further down in the same article:
> *Usually* the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions.
> As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations.
Putting it together:
1. The classic western geometry was predominantly non-synthetic, with the term “synthetic” being introduced much later in order to differentiate pure geometry from Analytic/algebraic forms
2. Use of formulae is outside the scope of pure geometry ala Euclidian style.
3. Distances and angles are geometric shapes. Trigonometry using algebra is part of analytic geometry.
4. Indian mathematicians used formulae and algebraic manipulations to arrive at geometric insights, including developing trigonometric identities and series.
5. Explicit use of coordinate system is not essential. It’s “usually” used.
6. Descartes didn’t invent the number line. He only proposed a coordinate system. Number line itself was a much later development. Descartes’ coordinates were a lot more abstract.
If using the modern coordinate system is your criteria for analytic geometry, when was it finally established?
What am I missing? Genuinely curious and happy to be completely wrong here.
Despite your attempt to obfuscate the issue with your wall of text, the simple fact of the matter is that you didn't give a single example of analytic geometry (i.e. coordinate geometry), which is based on the explicit use of a coordinate system.
> Use of formulae is outside the scope of pure geometry ala Euclidian style.
Like Heron's formula?
> Trigonometry using algebra is part of analytic geometry.
Yes, a part of analytic geometry. Distances and angles are also a part of analytic geometry.
> Explicit use of coordinate system is not essential.
Yes it is. That's literally what analytic geometry (i.e. coordinate geometry) means.
You use coordinate and analytic interchangeably but coordinate system is not a requirement nor equivalent to analytic geometry. It doesn’t even relate to mathematical analysis. Analysis, to the extent that it’s series and limits, also existed in India before Descartes.
Herons formula doesn’t count because it’s the end result. Not a means to an end. You don’t need to go to Heron for that. Area of a square also has a formula and that would have made your point.
Using trigonometric identities to derive area of cyclic quadrilateral is different as algebraic equations are means and not the end.
Again, analytic literally doesn’t mean “coordinate”. It only usually does.
Also it wasn’t a wall of text to obfuscate. I sincerely laid out the full argument the best I could. Half the wall is Wikipedia quotes. The other half was itemized to make it easy to parse. But sure whatever floats your boat mate.
A horse walks into a bar. The bartender asks the horse if they're an alcoholic considering all the bars they frequents, to which the horse replies "I think not!". POOF! The horse disappears.
This is the point in time when all the philosophy students in the audience begin to giggle, as they are familiar with the philosophical proposition of "cogito ergo sum", or "I think, therefore, I am".
But to explain the concept beforehand would be putting Descartes before the horse.
27 comments
[ 4.9 ms ] story [ 69.7 ms ] threadEdit: I found a draft from 2018, but the author asks for it not to be cited. It was published earlier this year, as: "Free Speech Skepticism". Susan J. Brison. Kennedy Institute of Ethics Journal, Volume 31, Number 2, June 2021
Ummm, never really thought about fingering a philosopher before
This kind of rhetorical gambit really bugs me. So if I find Gadberry unconvincing (or possibly even full of it) I'm close-minded and inattentive, am I?
It is that process that made it possible for elementary schoolchildren to reasonably be expected to cope with algebra and world history, topics largely limited to the old world's most brilliant academics.
Decartes was in the thick of that transition.
Resistance to zero and negative numbers had to evaporate once the number line was understood.
Source?
The key insight in analytic geometry is that geometric shapes and algebra are facets of the same underlying concept. This allowed Descartes to develop coordinate system. This in turn allowed development of calculus and everything else. This was a striking discovery in the western world.
The practice of using equations for geometry however was very common in India. Brahmagupta was the first to have a formula for area of cyclic quadrilateral in 6th century [2]. Notably the proof used trigonometry. Earlier similar result in West is Heron’s formula for the area of triangle. However the proof doesn’t use any algebra.
Bhaskara had devised ideas of differential calculus by 11th century [2]. A concrete example of this is the use of sine and cosine series in Kerala School of Mathematics [3].
This is really just the tip of the iceberg because majority of Indian mathematics is simply lost. What remains is poorly understood. But when reliable translations are available you can see that idea of using geometry and algebra interchangeably was common.
Maybe I’m biased. Curious to know what others make of this
[1] https://en.wikipedia.org/wiki/Brahmagupta%27s_formula [2] https://en.wikipedia.org/wiki/Bh%C4%81skara_II [3] https://en.wikipedia.org/wiki/Kerala_school_of_astronomy_and...
That's not analytic geometry.
> Bhaskara had devised ideas of differential calculus by 11th century [2]. A concrete example of this is the use of sine and cosine series in Kerala School of Mathematics [3].
I don't think these count as analytic geometry.
Analytic geometry has a specific meaning: https://en.wikipedia.org/wiki/Analytic_geometry
Anyway I’m not going to seriously discuss with you because it’s clear from your post history that your replies are avalanches of pedantry. If you think it has a specific meaning, elaborate it and identify the point of divergence or express what your view is. I don’t have an obligation to convince you.
Distances and angles have been used by the Greeks since ancient times. By themselves they don't constitute analytic geometry.
> Describing geometric objects with equations in coordinate system is analytic geometry.
Yes, and the example you gave does not satisfy that definition.
Yes, I have a degree in it.
> And yes that is analytic geometry
No, the example you gave is not analytic geometry. Read the definition again. The explicit use of a coordinate system is essential. Algebra is not sufficient.
> Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates *or formulae*. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems.
> Only after the introduction of coordinate methods was there a reason to introduce the term "synthetic geometry" to distinguish this approach to geometry from other approaches.
From wiki of analytic geometry [2]:
> In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This *contrasts* with synthetic geometry.
I know you’re using this to dismiss my case but the first line of Wikipedia hardly a standard for definition. Further down in the same article:
> *Usually* the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and squares, often in two and sometimes three dimensions.
> As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations.
Putting it together:
1. The classic western geometry was predominantly non-synthetic, with the term “synthetic” being introduced much later in order to differentiate pure geometry from Analytic/algebraic forms
2. Use of formulae is outside the scope of pure geometry ala Euclidian style.
3. Distances and angles are geometric shapes. Trigonometry using algebra is part of analytic geometry.
4. Indian mathematicians used formulae and algebraic manipulations to arrive at geometric insights, including developing trigonometric identities and series.
5. Explicit use of coordinate system is not essential. It’s “usually” used.
6. Descartes didn’t invent the number line. He only proposed a coordinate system. Number line itself was a much later development. Descartes’ coordinates were a lot more abstract.
If using the modern coordinate system is your criteria for analytic geometry, when was it finally established?
What am I missing? Genuinely curious and happy to be completely wrong here.
[1] https://en.wikipedia.org/wiki/Synthetic_geometry [2] https://en.wikipedia.org/wiki/Analytic_geometry
> Use of formulae is outside the scope of pure geometry ala Euclidian style.
Like Heron's formula?
> Trigonometry using algebra is part of analytic geometry.
Yes, a part of analytic geometry. Distances and angles are also a part of analytic geometry.
> Explicit use of coordinate system is not essential.
Yes it is. That's literally what analytic geometry (i.e. coordinate geometry) means.
Herons formula doesn’t count because it’s the end result. Not a means to an end. You don’t need to go to Heron for that. Area of a square also has a formula and that would have made your point.
Using trigonometric identities to derive area of cyclic quadrilateral is different as algebraic equations are means and not the end.
Again, analytic literally doesn’t mean “coordinate”. It only usually does.
Also it wasn’t a wall of text to obfuscate. I sincerely laid out the full argument the best I could. Half the wall is Wikipedia quotes. The other half was itemized to make it easy to parse. But sure whatever floats your boat mate.
Yes it is. That's what analytic geometry means [0][1][2][3]. If you mean something different, use a different word.
> Herons formula doesn’t count because it’s the end result.
It came with a derivation that can be found in Heron's 60 AD book Metrica, and may have been known centuries earlier.
[0] https://www.merriam-webster.com/dictionary/analytic%20geomet...
[1] https://www.dictionary.com/browse/analytic-geometry
[2] https://www.collinsdictionary.com/dictionary/english/analyti...
[3] https://www.thefreedictionary.com/analytic+geometry
This is the point in time when all the philosophy students in the audience begin to giggle, as they are familiar with the philosophical proposition of "cogito ergo sum", or "I think, therefore, I am".
But to explain the concept beforehand would be putting Descartes before the horse.