Square circles aren't possible in Euclidean Geometry - so they are logically inconsistent there; but they are possible in other geometries:

Equip the plane with the the L1 norm; and draw the circle; and then stand back and look at it - it's a square circle.

Paradoxes come in many forms - zen koans, physical singularities, logical absurdities, reductios and so on.

Sometimes they say: this far, and no further; at other times they are pregnant with thought.

Here is another example: division by zero; not 0/5, which is well defined within the formal context of arithmetic; but 0/0, which is not - it's undefined, it could be any number - but this turns out to be unuseful and unusable.

But 0/0, when considered as dx/dy, is not - it's fruitful, being the formal concept of the calculus; but why do I say this - after all it's not how historically calculus was invented.

Mathematicians like closure: where all operations and moves are well defined; a point of exception, instead is a challenge; and often proves to be the site of a new idea.

Thus 0/0, can be considered a site of exception, which when pushed through generates a new site, of a different order - not the arithmetic, but the analytic; this may, on the face of it, seem strange, or bizarre - but consider another site of exception, by way of comparison: the square root of -1, whose proper answer is i, the imaginary - it generates geometry: the arcane diagram; it's very name signals the differance of degree that pushing past this site has provoked.

Proper points of exceptions in mathematics (as opposed to mere or illusionary such points), may be considered as sites, where several concepts fuse, in an alchemical act of the mathematical imagination, and reveal a hitherto hidden dimension of depth in the being that is mathematics.

Square circles aren't possible in Euclidean Geometry - so they are logically inconsistent there; but they are possible in other geometries:

Equip the plane with the the L1 norm; and draw the circle; and then stand back and look at it - it's a square circle.

Paradoxes come in many forms - zen koans, physical singularities, logical absurdities, reductios and so on.

Sometimes they say: this far, and no further; at other times they are pregnant with thought.

Here is another example: division by zero; not 0/5, which is well defined within the formal context of arithmetic; but 0/0, which is not - it's undefined, it could be any number - but this turns out to be unuseful and unusable.

But 0/0, when considered as dx/dy, is not - it's fruitful, being the formal concept of the calculus; but why do I say this - after all it's not how historically calculus was invented.

Mathematicians like closure: where all operations and moves are well defined; a point of exception, instead is a challenge; and often proves to be the site of a new idea.

Thus 0/0, can be considered a site of exception, which when pushed through generates a new site, of a different order - not the arithmetic, but the analytic; this may, on the face of it, seem strange, or bizarre - but consider another site of exception, by way of comparison: the square root of -1, whose proper answer is i, the imaginary - it generates geometry: the argand diagram; it's very name signals the differance of degree that pushing past this site has provoked.

Proper points of exceptions in mathematics (as opposed to mere or illusionary such points), may be considered as sites, where several concepts fuse, in an alchemical act of the mathematical imagination, and reveal a hitherto hidden dimension of depth in the being that is mathematics.

Square circles aren't possible in Euclidean Geometry - so they are logically inconsistent there; but they are possible in other geometries:

Equip the plane with the the L1 norm; and draw the circle; and then stand back and look at it - it's a square circle.

Paradoxes come in many forms - zen koans, physical singularities, logical absurdities, reductios and so on.

Sometimes they say: this far, and no further; at other times they are pregnant with thought.

Square circles aren't possible in Euclidean Geometry - so they are logically inconsistent there; but they are possible in other geometries:

Equip the plane with the the L1 norm; and draw the circle; and then stand back and look at it - it's a square circle.

Paradoxes come in many forms - zen koans, physical singularities, logical absurdities, reductios and so on.

Sometimes they say: this far, and no further; at other times they are pregnant with thought.

Here is another example: division by zero; not 0/5, which is well defined within the formal context of arithmetic; but 0/0, which is not - it's undefined, it could be any number - but this turns out to be unuseful and unusable.

But 0/0, when considered as dx/dy, is not - it's fruitful, being the formal concept of the calculus; but why do I say this - after all it's not how historically calculus was invented.

Mathematicians like closure: where all operations and moves are well defined; a point of exception, instead is a challenge; and often proves to be the site of a new idea.

Thus 0/0, can be considered a site of exception, which when pushed through generates a new site, of a different order - not the arithmetic, but the analytic; this may, on the face of it, seem strange, or bizarre - but consider another site of exception, by way of comparison: the square root of -1, whose proper answer is i, the imaginary - it generates geometry: the arcane diagram; it's very name signals the differance of degree that pushing past this site has provoked.

Proper points of exceptions in mathematics (as opposed to mere or illusionary such points), may be considered as sites, where several concepts fuse, in an alchemical act of the mathematical imagination, and reveal a hitherto hidden dimension of depth in the being that is mathematics.