Truth, Mathematics, and Models (Part 8 in a series on the scientific method)

(Part 8 In A series on THE scientist method)

In THE last Payment I advance A assumption about What truth East, which East has say, I suggested A path has explain THE wide consensus that appears has exist about truth.  That explanation was: there East A objective reality "out there", And TRUE declarations are those that In a few sense match has THE real State of business In that objective reality.  This was problematic For declarations as, "Gandalf" was A Wizard" because Gandalf doesn't In fact exist In objective reality, but that was accounted for For by observe that THE real meanings of sentences In natural LANGUAGES often go beyond THE literal.

But there East A appearance of truth that East Stronger has account For, And which would be appear has First of all a look has be A serious default of My theory: mathematics.  Most people would be consider, For example, “1+1=2” Or "7 East prime" has be TRUE despite THE do that It is hard has map those Notions on Nothing In objective reality.  I can to show You "A apple" Or "A sheep", but showing You "A" East Stronger.  THE entire point of Numbers, And mathematics And logic In general, East has abstract far Since THE physical.  Numbers qua Numbers TO DO not exist In THE real world.  They are pure abstraction, Or has less they are supposed has be.  Mathematical truth East specifically destined No has be quota on Nothing In THE physical world, And SO he would be seem that My theory of truth failed has capture mathematical truth.

A few philosophers And religious apologists claim that he East SO impossible has ground mathematical truth In objective reality, that THE existence of mathematical truth requires something more, a few ethereal kingdom of Platonic ideals Or THE Spirit of God, has be THE source of such truths.  It is A plausible argument, but It is fake.  Mathematical truth can be Understood purely In terms of objective reality.  Specifically, mathematics can be seen as THE study of possible models of objective reality.  In This Payment I will explain What I mean by that.

There are A plot of different examples of (What East considered has be) "mathematical truth" but to leave Me to start with THE most basic: elementary arithmetic.  These include banal truths about Numbers, things as "two more three equals five" Or "Seven East prime."  He would be seem has First of all a look that Numbers used In This path don't do it refer has Nothing In objective reality.  I can to show You two of something but I can't to show You "two" In isolation.

There East A easy answer has This: Numbers In common use are not names, they are Adjectives.  THE reason I can't to show You "two" without showing You two of something East THE even reason I can't to show You "Green" unless I to show You A Green thing.  Adjectives to have has be related has names has be exposed, but that doesn't mean that "Green" do not exist In objective reality.  He do, It is just not A thing.  Green East A color, which East A property of things, but he East not himself A thing.  Likewise, "two" East not thing, he East A quantity, which East A property of (collections of) things.  And THE reason that two more three equals five East that if I to have two things And I put them together with three other things THE result East A quantity of things has which English speakers attach THE label "five".  Likewise "Seven East prime" can be Understood has mean that if I to have A quantity of things has which English speakers attach THE label "Seven" I can't organize those things In A complete, regular rectangular grid In any of them path other that THE degenerate case of putting them all In A line.

But This explanation failed For direct extensions of THE natural Numbers, as negative Numbers Or irrational Numbers Or imaginary Numbers.  I can to show You two apples, And I can explain addition And substraction In terms of putting groups of apples together And socket apples far, but only For positive Numbers.  I can't explain "three less five equals negative two" In terms of departure with three apples And socket five far because that East just not physically possible.  Likewise I can't to show You A square with A negative area, And SO I can't explain THE square roots of negative Numbers In terms of Nothing physical (has less not easily).

There are two more case Or THE numbers are adjectives theory failed.  THE First of all East truths that involve generalizations on Numbers as "There are A infinity number of first."  That can't be explain In terms of appropriate...