: : There could be a five-sided square. I've never seen one; and according to my understanding they don't exist; but I'm not the ultimate authority. Sorry, Stuart, I thought I'd made myself pretty clear there.: : I can never say 'there are no five-sided squares' without invoking synthetic a priori assumptions or definitions (i.e. what the definition of a 'square' is.)
: Good grief Farinata, sometimes I think you scientists deconstruct too much for your own good! What is the definition of a square?
Well, it depends upon the geometry you're using. According to strict Euclidean geometry, a square is a rectangle with four equal sides and four internal right angles (90° angles). However, in Euclidean space, an equilateral triangle has 3 60° internal angles (adding up to 180°) - but map this onto non-Euclidean (positively curved) space and you can have a triangle with three internal right angles; adding up to 270°.
(Try visualising this by projecting a big triangle onto the Earth; one point is at the North Pole; the other two are on the equator at 0° long. and 90° long. - you can see that lines of longitude intersect the equator at 90°; and the angle between them at the North Pole is also 90°...); the Earth is a curved surface and thus the laws of Euclidean geometry do not apply exactly; at small scales they're 'good enough'; at large scale like the one in this example, they fall down badly.
As such, the statement "the sum of the internal angles of the triangle add up to 180°" (that we all got taught at school) only applies for the special case of Euclidean (flat) space.
Of course, most people don't examine the definition that rigorously; ask them what a square is and they will draw something with four sides and four right angles; however, that definition only applies to 'a square under Euclidean space'; not to the entire set of all possible spatial geometries; under a negatively curved space, you could have a square with 4 60° angles.
: A rectangle with all four sides equal. Who created this definition? People did!
Euclid did.
: There can be no five-sided squares yet undiscovered because by man-made definition they would no longer be squares!
Only according to the Euclidean definition; which is a helpful approximation rather than The Ultimate Truth.
There might indeed be a five-sided square; but it would require non-standard definitions of the terms 'square' or 'five' or 'side'.
The point I was making to Stuart was that you can never make the blanket statement "there are no five-sided squares" - to do this would require complete knowledge of all the possible Universes in their entirety. You can only say "there are no five-sided squares according to my definition of the Universe and my experience of that Universe."
: Could there be semolina spaghetti made from spam?
Yes, although I've not experienced it personally.
: Could there be a toaster which makes bread soggy, not crisp?
Yes - fill your toaster with water ;)
: Allow humanity the luxury of defining things and abiding by those definitions; it's one thing to be a skeptic, but quite another to be a naysayer for the sake of naying itself. ("Naying." (v) To say nay).
This is fine as long as you have one common definition of what x is.
Taking my triangular example above, one person could say 'the internal angles of a triangle add up to 180°' - another could say 'aha! but the internal angles of *this* triangle add up to 270°' - and they'd both be absolutely correct; they have merely failed to state the caveats and implicit assumptions in their model; the spatial geometry that they were using.
The statements 'there are five-sided squares' and 'there are no five-sided squares' fall into exactly the same category; they both fail to put the qualifying assumptions.
If you included, say, space-time, then a cube would have four dimensions. Unless you say this specifically, anyone you ask will say quite firmly that a cube has three; length, breadth and height.
(Hardened physicists will probably say a number between 4 and 'possibly infinite' for the number of dimensions a cube has...)
As such, my original statement to Stuart was considered; I've never seen a five-sided square; my physical perceptions and model of the Universe specifically prohibit five-sided squares. But I can't in all integrity say that they do not exist; merely that the physical evidence I've seen makes them impossible to perceive physically.
Which is where we come down to reasonable belief; it is more reasonable to most people to agree with something that is backed up by all the physical evidence but not made certain either way by experimentation (e.g. that there are dodos left alive somewhere) than it is to believe something that flies in the face of all our physical evidence (that five-sided squares exist and frolic in fields of clover) - because the only evidence that we have to support or refute such an idea is our own physical perception.
I know it's pretty abstruse and overly picky, but it's best to be clear about this; I can no more say 'God does (or doesn't) exist' than I can say 'Five-sided squares exist (or don't exist)'. However, in the absence of any firm data either way, I go with what I've got; my physical perceptions and models of the Universe; which is why I feel that my position is reasonable; it is at least founded on something I can demonstrate and replicate under experimentation.
Farinata.
