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As usual your questions cut through to the very bone. I shall try to articulate an answer (or at least a sketch thereof), knowing all too well that this is impossible in full.

I believe the method is universal: suppose you have several analytic expressions and you surmise they are in fact the same, then IF you can prove explicitly that this is indeed the case, you form an equivalence class, and you say that these two versions are different instances of presentation of one "conceptual" object.

But I have already talked long without actually addressing directly your question: is analytic /synthetic used for anything but geometry?

The answer is here:

https://ncatlab.org/nlab/show/synthetic+mathematics

As you can see, there are several flavors available, such as synthetic topology, probability, domain theory, etc. This effort is by no mean new, but it is true that the categorical approach has once again emphasized the synthetic over the analytical

The answer is here:

https://ncatlab.org/nlab/show/synthetic+mathematics

As you can see, there are several flavors available, such as synthetic topology, probability, domain theory, etc. This effort is by no mean new, but it is true that the categorical approach has once again emphasized the synthetic over the analytical

SOME THOUGHTS AND A BIT OF BACKGROUND

As usual your questions cut through to the very bone. I shall try to articulate an answer (or at least a sketch thereof), knowing all too well that this is impossible in full.

I believe the method is universal: suppose you have several analytic expressions and you surmise they are in fact the same, then IF you can prove explicitly that this is indeed the case, you form an equivalence class, and you say that these two versions are different instances of presentation of one "conceptual" object.

As usual your questions go to the very bone. I shall try to articulate an answer (or at least a sketch thereof), knowing all too well that this is impossible in full.

The two terms, Analysis and Synthesis, go well before Kant, to the very beginning of western thought ( they somehow appeared in Aristoteles, for instance in his Analytica Priora, but he probably incorporated previous knowledge).

From the etimolopgical dictionary .

In modern day we could say that math is divided into two (and entwined) camps, analytics, and synthetic (in Lawvere's terminology the second one is called CONCEPTUAL). One would be tempted to replace them with constructive vs platonic, and perhaps one would not go too far off, but with many caveats.

The examples abound: for instance, one can define a smooth manifold intrinsically, or present it with charts.

Now, what is interesting is the interplay between these two sides: how to pass from the synthetic to the analytic viewpoint and conversely?

I believe the method is universal: suppose you have several analytic expressions and you surmise they are the same, then IF you can prove explicitly that this is indeed the case, you form an equivalence class, and you say that these two versions are different instances of presentation of one "conceptual" object.

But I have already talked long without actually addressing your question: is analytic /synthetic used for anything but geometry?

As you can see, there are several flavors available, such as synthetic topology, probability, domain theory, etc.

PS Descartes is the official father of analytic geometry, even though some Greek anticipated him.

What is intriguing is that Descartes is in a way the father of both modern philosophy and mathematics. In his Discourse of the Method he elaborated his universal way of reasoning, which distinguished 4 phases, 2 of them being Analysis and Synthesis. It is not too much of a stretch thinking that his analytic geometry is an application of his own method (of which he was very proud, and rightly so)

PPS it is not by chance that Lawvere has written (with Schanuel) Conceptual Mathematics. Category Theory (including higher cats) is so far perhaps the best tool we have to treat Synthetic Math, though by no means the only one. That is both its greatness and its limit: it captures the invariant side of math, but of course it hides the other one (think of groups for instance: to study the cat of Groups tells you all you need to know of general groups, but if I want to study say SU(5) I need to calculate....

As usual your questions cut through to the very bone. I shall try to articulate an answer (or at least a sketch thereof), knowing all too well that this is impossible in full.

The two terms, Analysis and Synthesis, go back well before Kant, to the very beginning of western thought ( they somehow appeared in Aristoteles, for instance in his Analytica Priora, ie the first formalization of logic, but he probably incorporated previous knowledge from various sources).

It is worth visiting etimological dictionary:

In modern day we could say that math is like an acrobat balancing itself between two (yet entwined) sides, the analytic one, and the synthetic (in Lawvere's terminology the second is called CONCEPTUAL). 

One would be tempted to replace them with constructive vs platonic, and perhaps one would not go too far off, but with many caveats.

The examples abound: for instance, one can define a smooth manifold intrinsically, or present it with explicit charts. A group is an instance of an abstract group, but Combinatorial Group Theory studies concrete groups generated by a set of elements and its relations. You can pick your field, and you almost invariably see that it straddles between an analytical approach and a synthetic one (though the mixture varies...)

Now, what I find extremely interesting is the interplay between these two sides: how to pass from the synthetic to the analytic viewpoint and conversely?

I believe the method is universal: suppose you have several analytic expressions and you surmise they are in fact the same, then IF you can prove explicitly that this is indeed the case, you form an equivalence class, and you say that these two versions are different instances of presentation of one "conceptual" object.

But I have already talked long without actually addressing directly your question: is analytic /synthetic used for anything but geometry?

As you can see, there are several flavors available, such as synthetic topology, probability, domain theory, etc. This effort is by no mean new, but it is true that the categorical approach has once again emphasized the synthetic over the analytical

PS Descartes is the official father of analytic geometry, even though some Greek (forgot his name) anticipated him.

What is intriguing is that Descartes is in a way the father of both modern philosophy and mathematics. In his Discourse of the Method he elaborated his universal way of reasoning, which distinguished 4 phases, 2 of them being Analysis and Synthesis. It is not too much of a stretch to think that his analytic geometry is an application of his own philosophical method (of which he was very proud, and rightly so) . But, not knowing enough of this story, I could be wrong: it could be that the mathematician arrived before the philosopher. Either way, there is no doubt that this question carries a great relevance in both philosophy and mathematics.

PPS it is not by chance that Bill Lawvere has written (with Schanuel) a book called Conceptual Mathematics. Category Theory (including higher cats) is so far perhaps the best tool we have to treat Synthetic Math in full generality, though by no means the only one. That is both its greatness and its inherent limit: it captures the invariant side of math, but of course it hides the other one (think of groups for instance: to study the cat of Groups tells you all you need to know of general groups, but if I want to study say SU(5) I need to calculate, there is no way round it....

As usual your questions go to the very bone. I shall try to articulate an answer (or at least a sketch thereof), knowing all too well that this is impossible in full.

The two terms, Analysis and Synthesis, go well before Kant, to the very beginning of western thought ( they somehow appeared in Aristoteles, for instance in his Analytica Priora, but he probably incorporated previous knowledge).

From the etimolopgical dictionary .

Analysis, circa 1580s, "resolution of anything complex into simple elements" (opposite of synthesis), from Medieval Latin analysis (15c.), from Greek analysis "solution of a problem by analysis," literally "a breaking up, a loosening, releasing," noun of action from analyein "unloose, release, set free; to loose a ship from its moorings," in Aristotle, "to analyze," from ana "up, back, throughout" (see ana-) + lysis "a loosening," from lyein "to unfasten" (from PIE root *leu- "to loosen, divide, cut apart").

**So, Analysis basically means to solve some concept into its simpler
constituents, whereas Synthesis is the opposite direction.** 

In modern day we could say that math is divided into two (and entwined) camps, analytics, and synthetic (in Lawvere's terminology the second one is called CONCEPTUAL). One would be tempted to replace them with constructive vs platonic, and perhaps one would not go too far off, but with many caveats.

The examples abound: for instance, one can define a smooth manifold intrinsecally, or present it with charts.

Now, what is interesting is the interplay between these two sides: how to pass from the synthetic to the analytic viewpoint and conversely?

I believe the method is universal: suppose you have several analytic expressions and you surmise they are the same, then IF you can prove explicitly that this is indeed the case, you form an equivalence class, and you say that these two versions are different instances of presentation of one "conceptual" object.

In other words, the manifold is the concept which admits several chart presentations.

PS Descartes is the official father of analytic geometry, even though some Greek anticipated him.

What is intriguing is that Descartes is in a way the father of both modern philosophy and mathematics. In his Discourse of the Method he elaborated his universal way of reasoning, which distinguished 4 phases, 2 of them being Analysis and Synthesis. It is not too much of a stretch thinking that his analytic geometry is an application of his own method (of which he was very proud, and rightly so)

PPS it is not by chance that Lawvere has written (with Schanuel) Conceptual Mathematics. Category Theory (including higher cats) is so far perhaps the best tool we have to treat Synthetic Math, though by no means the only one. That is both its greatness and its limit: it captures the invariant side of math, but of course it hides the other one (think of groups for instance: to study the cat of Groups tells you all you need to know of general groups, but if I want to study say SU(5) I need to calculate....

As usual your questions go to the very bone. I shall try to articulate an answer (or at least a sketch thereof), knowing all too well that this is impossible in full.

The two terms, Analysis and Synthesis, go well before Kant, to the very beginning of western thought ( they somehow appeared in Aristoteles, for instance in his Analytica Priora, but he probably incorporated previous knowledge).

From the etimolopgical dictionary .

Analysis, circa 1580s, "resolution of anything complex into simple elements" (opposite of synthesis), from Medieval Latin analysis (15c.), from Greek analysis "solution of a problem by analysis," literally "a breaking up, a loosening, releasing," noun of action from analyein "unloose, release, set free; to loose a ship from its moorings," in Aristotle, "to analyze," from ana "up, back, throughout" (see ana-) + lysis "a loosening," from lyein "to unfasten" (from PIE root *leu- "to loosen, divide, cut apart").

**So, Analysis basically means to solve some concept into its simpler
constituents, whereas Synthesis is the opposite direction.** 

In modern day we could say that math is divided into two (and entwined) camps, analytics, and synthetic (in Lawvere's terminology the second one is called CONCEPTUAL). One would be tempted to replace them with constructive vs platonic, and perhaps one would not go too far off, but with many caveats.

The examples abound: for instance, one can define a smooth manifold intrinsically, or present it with charts.

Now, what is interesting is the interplay between these two sides: how to pass from the synthetic to the analytic viewpoint and conversely?

I believe the method is universal: suppose you have several analytic expressions and you surmise they are the same, then IF you can prove explicitly that this is indeed the case, you form an equivalence class, and you say that these two versions are different instances of presentation of one "conceptual" object.

But I have already talked long without actually addressing your question: is analytic /synthetic used for anything but geometry?

The answer is here:

https://ncatlab.org/nlab/show/synthetic+mathematics

As you can see, there are several flavors available, such as synthetic topology, probability, domain theory, etc.

PS Descartes is the official father of analytic geometry, even though some Greek anticipated him.

What is intriguing is that Descartes is in a way the father of both modern philosophy and mathematics. In his Discourse of the Method he elaborated his universal way of reasoning, which distinguished 4 phases, 2 of them being Analysis and Synthesis. It is not too much of a stretch thinking that his analytic geometry is an application of his own method (of which he was very proud, and rightly so)

PPS it is not by chance that Lawvere has written (with Schanuel) Conceptual Mathematics. Category Theory (including higher cats) is so far perhaps the best tool we have to treat Synthetic Math, though by no means the only one. That is both its greatness and its limit: it captures the invariant side of math, but of course it hides the other one (think of groups for instance: to study the cat of Groups tells you all you need to know of general groups, but if I want to study say SU(5) I need to calculate....