further the paper goes on to prove
1+1=1
contradicting
1+1=2
thus mathematics ends in contradiction

"1+1=2"

Also:
1+1=11 (side by side)

1+1=4 (rotate one '1' counter-clockwise)

1+1=8 (2 cats produce 6 kittens)

The outside world of shapes and forms is full of contradictions because it is not (and cannot be) well defined, in fact it is an illusion.

However, mathematical objects and the operations that apply to them are well defined so the results of these operations are always the same. Similarly, a geometric point has no shape, form or size, so nobody can draw a geometric point in the world that is outside mathematics.

You seem to mix up the mathematical objects with their representation: a "number" is not a mathematical object, it only represents a mathematical object. Mathematical operations can only be applied to mathematical objects, thus:

1 number + 1 number = 1 number

cannot be reduced to "1+1=1" since you cannot divide (both sides) by "number", which is not a mathematical object.

//further the paper goes on to prove
1+1=1//

Not in any conventional sense of the word 'prove'.

Get off the crack, it's not doing you any favours.

The fact is
The paper proves

1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way

1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction

and

1+1=2
and
1+1=1
1 number +1 number=1 number
or
1 heap of salt + 1 heap of salt= 1 heap of salt
thus maths ends in contradiction

Dear Sara,

One cannot perform arithmetic operations on "a heap of salt", only on well and axiomatically-defined numbers.

AND

0.999... IS an integer. Even the paper you presented does not claim that it isn't, how less so proves so.

when a finite number 1 = a non-finite number 0.999.., this simply demonstrates that one number can be represented in two (or more) different ways. In this case, one WAY OF REPRESENTATION is finite, the other not: that says nothing about the finiteness or otherwise of the actual number itself.

sara242,
The sum "1+1=1" is never right. We can make it right by multiplying each side be zero, but mathematicians are well aware of that, and multiplying both sides by zero is not treated as equivalent to multiplying them by another number because we know from observation that it makes things equal that would ordinarily be unequal. When an apparent contradiction is encountered, mathematics sets conditions to avoid it; the contradiction is resolved and maths keeps going. This is because mathematics is about truth - and the truth is that 1 is not equal to 2 even if you "prove" otherwise by multiplying by zero.

Zero isn't the only value which has to be treated differently in some mathematical laws. The same goes for infinity, and for minus infinity. NonSpecificNumber appears to be a fourth such value, though many would dispute its validity as a value. Likewise with a heap of salt.

Either way, mathematics sometimes has exceptions, but it never ever has contradictions (except as proof that something is FALSE). If you think you find a contradiction, it shows that either your reasoning is wrong or your assumptions are wrong (or both).

1+1=2 no matter how much you want it to be something else.