The Numbers Argument (Part 1)

According to the Philosophy of Numbers, there are two competing theories relative to the question “What are numbers, beyond their arithmetically assigned values?”  Are they abstract mental projections or abstract mental objects? 

Nominalism - Numbers exist as abstract mental projections.  Nominalism says a mind or minds is/are required in order for numbers to exist.  

Platonism  - Numbers exist independently of our minds.  They exist outside of space time itself.  They are abstract mental objects.   Platonism agrees with Nominalism on the necessity of a mind or minds in order to underwrite the existence of numbers.  

Query: If numbers exist as artifacts of our own minds, then in what mind or minds do they inhere?   Do they inhere in some minds but not others?  All minds? 

Query: Numbers constitute an infinite set of abstract mental objects or projections.  How can a collection of finite minds underwrite each and every integer?   

Let’s take the number with the absolute value of “One.” It is finite, yet at the same time infinite insofar as it cannot exist without the concomitant existence of the infinite set of integers that compose the set from zero to one.   Our minds are finite.  How can an infinite set exist or subsist in a finite mind or set of minds? 

Atheists sometimes posit the Nominalist view.  Why? Because, so they say, there is no evidence for the existence of any minds other than our own.   

Any mind that underwrites an infinite set of abstract mental *objects* would exist outside of space time itself and necessarily be infinite and immutable, just like numbers.    Nominalism turns on the premise that our minds, which exist in space-time, underwrite the existing of numbers as abstract mental projections, ie numbers exist within space-time itself as mental projections of our own psyche. 

 

It’s also worth noting that consistent atheism conduces to physicalism.  In physicalism, parts of the brain (the cerebrum) & mind are essentially the same.   How can any integer exist in a finite physical construct since, in order inhere in that construct, every preceding integer (an infinite set) would, of necessity, have to inhere in the brain/mind in order for that integer to exist.  

Atheists and Theists agree that there was a time in which human and animal minds did not exist.   They also agree that numbers are artifacts of the mind, eg a mjnd is necessary to underwrite them. 

Atheists typically run to something like the Big Bang in order to underwrite the existence of the universe we know.   That requires the existence of matter and energy which have structure.     

Query; What evidence do atheists have that matter and energy are able to exist without the concomitant existence of numbers that underwrite structure itself?  None whatsoever.   They do this while arguing that theists have little or no evidence that God exists. 

Theism is at home with the Platonist concept of numbers.   Numbers exist outside of space time.  They require a mind in which to inhere as well as subsist.    Theism, in which God is defined as infinite, eternal, all-knowing, & immutable, can account for the existence of numbers.  It can also account for the infinite and immutable nature of numbers.   It can also account for the evidence that where structure exists, numbers exist, and where numbers exist, there is a mind that exists in order to underwrite them. 

Atheists must resort to statements like this: Structure is able to exist without the concomitant existence of math and numbers.  Where is the evidence for that statement?  There is none. 

Numbers also display the problem of the one and many: unity and diversity within the same phenomenon.   Think of the number 1 as an abstract mental phenomenon.   The *number* is just a raw idea.  It has no substance, essence, or esse (physical form).   The numeral “1” is not the number itself.  Rather it represents the number.   Then there is the word “one.”  The word is neither the numeral nor the number.  It is, however, the word that represents both the numeral and the number.    Unity is expressed diversely - number, numeral, word — and each of these is abstract, an idea that expresses the same concept. 

Here is another instance of the problem: particle, wave, standing wave.   Sometimes atheists will point to these three as the answer or ground to the problem of the one & many.  That isn’t correct.  Why?  Because these physical phenomena also have measurements at the abstract level.   The particle, wave, standing wave answer is just an instance of the problem.   Particles, which are inanimate, physical, & finite, cannot concomitantly account for the existence of numbers which everybody agrees require a mind or mind to underwrite them.   

Query: What phenomenon can account for these issues related to infinity, knowledge, abstractions, numbers, and the problem of the one and many?  The existence of a mind that exists apart from our own, which is capable of grounding these observed phenomena - a mind that is both unity and diversity at the same time, and that is infinite and immutable as well.   

Those are qualities that fit what theologians describe as God’s mind.  For example, the Westminster Catechisms define God as a (triune) spirit (without physical, temporal substance, essence, esse) who is infinite, eternal, and immutable in being, wisdom, power, justice, goodness, holiness, and truth. 

God bless us all, each & every one.  Go & sin no more.