Hobbes' Choice wrote:
There is a difference between a real thing and an idea. If this were not the case then anything I dream would have material existence.
Maths is a system of ideas that have been developed to help us describe the universe. But most of its postulates are not real in any sense.
No two apples are the same, so the equation 1+1=2 is only an approximation. The two apples occupy different places in space, cannot have the same mass, vary in temperature... and so on.
There are no straight lines in nature, no perfect spheres, there are no integers. irrational numbers are - well irresolvable - and yet there are thought to pertain to things which are. Even Euclid's Fifth postulate is not true in reality.
So give logic and maths its correct place. They offer analytically true statements, that work within their own assumptions and postulates systems, but only approximate reality when applied to the real world.
Throughout you have offered an idea of maths as truth. You might just as well have said that there exists a world that only has 2 dimensions. Such an idea has been expressed in Flatland, but it all examples, reality demands thickness.
You haven't answered how using any essential thing like reasoning within any scientific argument can have any validity if the very validity of this reasoning is neither provable nor disprovable. This only adds confusion when you get to the premise in the method that states that all theories must be disprovable. Up front it suggests an odd mindset that appears to approve of uncertainty and an irrational means to proceed fairly. No wonder religions still persist elsewhere. But maybe this was done on purpose?
"No two apples are the same"? I already explained how we define things based on demonstrating similarity of ALL things to one class while distinguishing their essential differences making the member unique.
Let any letter represent objects here. In fact, let them only represent themselves as the letters they are:
1. TT ...........a set of two T's.
2. hh ............a set of two h's.
3. yy ............a set of two y's
induced general form of the above observational samples denoted:
C. xx ............a set of two x's such that 'x' stands for any letters in the domain = {T, h, y}. Let this domain be symbolized or assigned as D1.
Then let us call the above generalization "
A set of any two things given D1."
This is a form or generalization of specific things in reality that denote what 'Two' means. If you think only the initial things are real but not the concluding generalization they are based upon, then you have to always denote directly everything uniquely everywhere.
The domain, D1, above demonstrates the 'differences' between its members by both showing all options and limiting it to that domain. In reality, with regards to an infinite set, we add an element that stands as any variable defined. For here, this might be the class domain of all available letters one can use.
C above is real too but acts as a form into which you can place variable elements into.
If you question the real nature of this, you can revert this by declaring it conditionally true while maintaining initial experiments
1, 2, and 3 as real and what follows predictably. Any confirmation of what you think is true by the pattern already has to be true by the general form of it. And since you cannot have
C be both true and false of the same thing, either you have to abandon it or 'fix' it by providing a new dimension where it can exist. Thus,
C is always true regardless at least somewhere.
This proves that at least one 'form' or 'idea' exists independent of other normal everyday objects. This is a sample of the 'laws' of which science refers to when replacing the meaning of these symbols above with sensory data (observations) rather than mere data from memory. The external world is variable, and thus only equal in value to internal memory by their common means to be defined as real.
Two apples are the same to one another by what makes the definition of "apple" remain constant. If one apple happens to be bigger than the other, are you supposed to say I have one big-apple and one non-big-apple? What I mean is am I not allowed to refer to a common unit of measure, an "apple" where the definition serves to define things both similarly and differently?
I hope this helps convince you and Leo better but am beginning to doubt that if one cannot accept logic as being real, I am likely wasting any effort. And I won't reduce my argument to an appeal to a
Chewbacca Defence although I think this likely has more convincing power here by the way people think.