On Plato, the Early Church, and Modern Science: An Eclectic Meditation

November 30, 2004 | Jean F. Drew

[ckilmer]

The Real Number System

The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers.

Natural Numbers

or “Counting Numbers”

1, 2, 3, 4, 5, . . .

• The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.

At some point, the idea of “zero” came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus the number zero the whole numbers.

Whole Numbers

Natural Numbers together with “zero”

0, 1, 2, 3, 4, 5, . . .

 

About the Number Zero

What is zero? Is it a number? How can the number of nothing be a number? Is zero nothing, or is it something?

Well, before this starts to sound like a Zen koan, let’s look at how we use the numeral “0.” Arab and Indian scholars were the first to use zero to develop the place-value number system that we use today. When we write a number, we use only the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. These numerals can stand for ones, tens, hundreds, or whatever depending on their position in the number. In order for this to work, we have to have a way to mark an empty place in a number, or the place values won’t come out right. This is what the numeral “0” does. Think of it as an empty container, signifying that that place is empty. For example, the number 302 has 3 hundreds, no tens, and 2 ones.

So is zero a number? Well, that is a matter of definition, but in mathematics we tend to call it a duck if it acts like a duck, or at least if it’s behavior is mostly duck-like. The number zero obeys most of the same rules of arithmetic that ordinary numbers do, so we call it a number. It is a rather special number, though, because it doesn’t quite obey all the same laws as other numbers—you can’t divide by zero, for example.

Note for math purists: In the strict axiomatic field development of the real numbers, both 0 and 1 are singled out for special treatment. Zero is the additive identity, because adding zero to a number does not change the number. Similarly, 1 is the multiplicative identity because multiplying a number by 1 does not change it.

 

 

Even more abstract than zero is the idea of negative numbers. If, in addition to not having any sheep, the farmer owes someone 3 sheep, you could say that the number of sheep that the farmer owns is negative 3. It took longer for the idea of negative numbers to be accepted, but eventually they came to be seen as something we could call “numbers.” The expanded set of numbers that we get by including negative versions of the counting numbers is called the integers.

Integers

Whole numbers plus negatives

. . . –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .

 

About Negative Numbers

How can you have less than zero? Well, do you have a checking account? Having less than zero means that you have to add some to it just to get it up to zero. And if you take more out of it, it will be even further less than zero, meaning that you will have to add even more just to get it up to zero.

The strict mathematical definition goes something like this:

For every real number n, there exists its opposite, denoted – n, such that the sum of n and – n is zero, or

n + (– n) = 0

Note that the negative sign in front of a number is part of the symbol for that number: The symbol “–3” is one object—it stands for “negative three,” the name of the number that is three units less than zero.

The number zero is its own opposite, and zero is considered to be neither negative nor positive.

Read the discussion of subtraction for more about the meanings of the symbol “–.”

 

 

The next generalization that we can make is to include the idea of fractions. While it is unlikely that a farmer owns a fractional number of sheep, many other things in real life are measured in fractions, like a half-cup of sugar. If we add fractions to the set of integers, we get the set of rational numbers.

Rational Numbers

All numbers of the form , where a and b are integers (but b cannot be zero)

Rational numbers include what we usually call fractions

• Notice that the word “rational” contains the word “ratio,” which should remind you of fractions.

 

The bottom of the fraction is called the denominator. Think of it as the denomination—it tells you what size fraction we are talking about: fourths, fifths, etc.

 

The top of the fraction is called the numerator. It tells you how many fourths, fifths, or whatever.

 

• RESTRICTION: The denominator cannot be zero! (But the numerator can)

If the numerator is zero, then the whole fraction is just equal to zero. If I have zero thirds or zero fourths, than I don’t have anything. However, it makes no sense at all to talk about a fraction measured in “zeroths.”

• Fractions can be numbers smaller than 1, like 1/2 or 3/4 (called proper fractions), or they can be numbers bigger than 1 (called improper fractions), like two-and-a-half, which we could also write as 5/2

All integers can also be thought of as rational numbers, with a denominator of 1:

This means that all the previous sets of numbers (natural numbers, whole numbers, and integers) are subsets of the rational numbers.

Now it might seem as though the set of rational numbers would cover every possible case, but that is not so. There are numbers that cannot be expressed as a fraction, and these numbers are called irrational because they are not rational.

Irrational Numbers

• Cannot be expressed as a ratio of integers.
• As decimals they never repeat or terminate (rationals always do one or the other)

Examples: 

	Rational (terminates)
	Rational (repeats)
	Rational (repeats)
	Rational (repeats)
	Irrational (never repeats or terminates)
	Irrational (never repeats or terminates)

 

More on Irrational Numbers

It might seem that the rational numbers would cover any possible number. After all, if I measure a length with a ruler, it is going to come out to some fraction—maybe 2 and 3/4 inches. Suppose I then measure it with more precision. I will get something like 2 and 5/8 inches, or maybe 2 and 23/32 inches. It seems that however close I look it is going to be some fraction. However, this is not always the case.

Imagine a line segment exactly one unit long:
Now draw another line one unit long, perpendicular to the first one, like this:
Now draw the diagonal connecting the two ends:

Congratulations! You have just drawn a length that cannot be measured by any rational number. According to the Pythagorean Theorem, the length of this diagonal is the square root of 2; that is, the number which when multiplied by itself gives 2.

According to my calculator,

But my calculator only stops at eleven decimal places because it can hold no more. This number actually goes on forever past the decimal point, without the pattern ever terminating or repeating.

This is because if the pattern ever stopped or repeated, you could write the number as a fraction—and it can be proven that the square root of 2 can never be written as

for any choice of integers for a and b. The proof of this was considered quite shocking when it was first demonstrated by the followers of Pythagoras 26 centuries ago.

 

The Real Numbers

• Rationals + Irrationals
• All points on the number line
• Or all possible distances on the number line

When we put the irrational numbers together with the rational numbers, we finally have the complete set of real numbers. Any number that represents an amount of something, such as a weight, a volume, or the distance between two points, will always be a real number. The following diagram illustrates the relationships of the sets that make up the real numbers.

An Ordered Set

The real numbers have the property that they are ordered, which means that given any two different numbers we can always say that one is greater or less than the other. A more formal way of saying this is:

For any two real numbers a and b, one and only one of the following three statements is true:

1.      a is less than b, (expressed as a < b)

2.      a is equal to b, (expressed as a = b)

3.      a is greater than b, (expressed as a > b)

The Number Line

The ordered nature of the real numbers lets us arrange them along a line (imagine that the line is made up of an infinite number of points all packed so closely together that they form a solid line). The points are ordered so that points to the right are greater than points to the left:

• Every real number corresponds to a distance on the number line, starting at the center (zero).
• Negative numbers represent distances to the left of zero, and positive numbers are distances to the right.
• The arrows on the end indicate that it keeps going forever in both directions.

Absolute Value 

When we want to talk about how “large” a number is without regard as to whether it is positive or negative, we use the absolute value function. The absolute value of a number is the distance from that number to the origin (zero) on the number line. That distance is always given as a non-negative number.

In short:

• If a number is positive (or zero), the absolute value function does nothing to it:
• If a number is negative, the absolute value function makes it positive:

WARNING: If there is arithmetic to do inside the absolute value sign, you must do it before taking the absolute value—the absolute value function acts on the result of whatever is inside it. For example, a common error is

   (WRONG)

The correct result is

 

[ckilmer]

this tutorial is for the benefit of me and anyone else struggling to follow the math in this conversation.

the link comes from here
http://www.jamesbrennan.org/algebra/numbers/real_number_system.htm

[Eastbound]

"historically the arian position, while dominant is arabia--is an anomaly in the west."

I'm not certain of your meaning, ckilmer. Can you re-phrase? Thank you.

[StJacques]

". . . So you can't ignore the hole -- for the "whole" of the doughnut is comprised of the thing we can see and the thing we cannot see. . . ."

So then there is such a thing as "doughnutness" by which we may only view a doughnut in the material world as a "contingent" example made possible by its "necessary" objective reality?

Ok, now we're getting somewhere.

[ckilmer]

I think the same analogy would describe how different people (beliefs) view the concept of God. Some view God as having only a single aspect. Others, who have moved from their position, view God as a Triunity -- each representing the whole.
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historically the arian position, while dominant is arabia--is an anomaly in the west.
/////////
I'm not certain of your meaning, ckilmer. Can you re-phrase? Thank you.
//////////////
I'm responding to the "who have moved from their position" part of your statement.

I'm not sure what that's about. So I just stated that historically trinitarianism has been the dominant theology in the west. It has only been since the enlightenment that monism has been more prevalent--as practiced by unitarians (-1750)& jehovah witnesses +-1900)officially and liberal protestants (+-1935) unoffically in the USA and most protestant churches in Europe since the mid 1800's. The caveat for European churches is that they have very rapidly gone from monism to extinction. In Berlin alone some 70 churches are up for sale this year because no one attends. Monism is what happens when Jesus is considered to be no longer equal to God the Father. That is he is considered a great teacher/philosopher/prophet and very very good. The Moslems think of Jesus as being a wise man/teacher/prophet/very very very good--but not God. They take the assertion that Jesus is God in the same way that the Jews of Jesus time took Jesus words--as blasphemy.

There is no proof. Rather I have heard it said that the form of christianity that Mohammed encountered was arianism. Arianism, Named after Arias of Alexandria Egypt-- was the third century heresy against which the council of Nicea ruled in +-325. I have learned in recent years that-- during the civil wars of the roman empire of the period--the Arians and the neoplatonists were fellow travelers. Similiarly, you'll find that starting from the late 1700's the neoplatonists in the philosophy depts in the west and the purveyors of "higher criticism" in the seminaries were also fellow travelors.

Interestingly, I heard a presentation last sunday by the second of a team of free church affiliated--Calvinist German Seminarians who want to set up calvinist church in downtown Berlin. It would be the first in more than 200 years. He has been training at Redeemer Presbyterian Church in NYC. Redeemer is the first Calvinist (conservative/trinitarian) Presbyterian Church to be set up in any downtown area in the USA in many decades. Many liberal presbyterian downtown churches in NYC are virtually abandoned but they are not as far along as the liberal churches in Berlin. The German church planter I talked with was Christian Nowatzky. His historical take was much similiar to the one I presented above.

here's the url for that group.
http://www.redeemer2.com/themovement/issues/2004/august/berlinprojekt.html

My impression is the guys have the right stuff. Its my prayer that it can be said of the Germans in the next decade or so "who have moved from their position" ...

[ckilmer]

So then there is such a thing as "doughnutness" by which we may only view a doughnut in the material world as a "contingent" example made possible by its "necessary" objective reality?
////////////////
there's probably a better ditty for this but this is the way I heard it.

matter tells space how to bend.
space tells matter how to move.
//////////////
so space is not nothing.

[ckilmer]

"historically the arian position, while dominant is arabia--is an anomaly in the west."
////////////

I'm not certain of your meaning, ckilmer. Can you re-phrase? Thank you.
///////////////
sorry for the long explanation. The better sentence than the one I posted above would require the preposition "in" rather than the verb "is"--so it would read--
"historically the arian position, while dominant in arabia--is an anomaly in the west."

Sorry too for leaving out the explanation of Arianism. The arian position or Arianism is named after Arius of Alexandria who in the early 300's AD, --said that Jesus was fully Man but Not fully God. This caused a dispute in the church which was resolved at the council of Nicea under Constantine's auspices --in favor of the trinitarians--who held that Jesus is fully God as well as fully Man. However, there was still another 100+ years or so of wars in the late roman empire to be fought in which this theological debate played a part. Civil wars on this scale in which theology placed such a central role would not come again to Europe until the latter half of the 1500's and then would run for the next 100 years or so. The climactic catholic/protestant wars were during the 30 years wars of 1619-1649. However, nothing was really settled. Everyone remained pissed and bitter. And the animosity between catholics and protestants did not really subside until both were eclipsed and diminished by secularists in the 20th century.

Finally, people who believe in arianism are, necessarily, monists. --since jesus is out of the godhead-- as well as the holy spirit.

[Eastbound]

Thank you for the replies, ckilmer. Bookmarking for tomorrow evening. Trying to photograph the new comet near Orion but having difficulty.

[ckilmer]

Back 350 years ago when Liebniz helped to invent the calculus, mathematicians were not resticted to the real number system. They also used numbers called infinitesimals which are so small that you can think of them as zero. To find the slope of the spaghetti curve, Leibniz shrank the vertical and horizontal sides left over from our original triangle until those sides became infinitesimals. So you can think of this as dividing zero by zero, i.e. 0/0. The amazing result was an equation that gave him a finite slope that varied for every point on the curved spaghetti. Wow!!!

About 100 years ago mathematicians became very uneasy about infinitesimals and dividing 0 by 0, so they threw out infinitesimals, and redefined calculus in terms of the concept of limits and real numbers only.

Now, when I first took calculus, I was lost. I didn't understand limits at all and found them confusing and cumbersome. I started studying the history of the development of calculus. Since Leibniz used infinitesimals, I began to think of calculus in terms of infinitesimals. A light bulb went on, and I began to solve calculus problems with ease. But I didn't dare to tell my instructor that I was secretly thinking about dividing infinitesimals instead of taking limits of real numbers.
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The length of the diagonal of a right triangle is an irrational number. Therefor even if the vertical and horizontal of the right trangle were set at zero the hypotonuse would still be an irrational number.

Are you saying that Leibnitz's equation produced a sequence of irrational infintessimals? This raises two questions. does this mean that the space between each irrational infintessimal is --irreducable. If so then perhaps the answer to the old question "how many angels can you get on the head of a pin" ... is--a finite number of irrational infintessimals. This would also answer why it is that if you tried to get from one to two by going half the distance and then half the distance again forever---that you would never get from one to two.

The reason is that the distance between 1 and 2 is a straight line.

If not then none of the above conclusions hold. But perhaps you could comment.

second question: why is a sequence of irrational infitessimals that describe a finite slope--so darn exciting. Is it because theres so many angels in one spot/curve or because there's no space between them.

never mind. I'm getting jazzed thinking about it too. And I don't even understand math. Now I think I understand how you were motivating betty boop to expand on her thinking.

so now that we know that space is not nothing...what is space.

I believe that liebnitz was Newton's competitor/contemporary. Newton was such a towering figure in the sciences that his religious writings are generally overlooked. He was a thorough going Arian--in the sense that he believed that Jesus was fully man but not really God. Because 200 years before Newton--the reformation coincided with the overturning of the Ptolemaic cosmology
in the early 1500's--I wonder whether the math and science that Newton produced didn't also the overthrow reformation theology. Or whether it was the man himself who set things in motion.

If the the clue to Newton's theology could be found in Newton's calculus--what number or calculus would you say it was. I don't know anything about Liebnitz beyond what I've learned in this thread. Perhaps his numbers have the clue.

[betty boop]

Sort of like discourse about Platonic 'forms'.... There's no 'there' there.

Are you saying that just because you can't "see" a thing (i.e., because it is not a datum of ordinary sense experience, and therefore is beyond the reach of direct observation and experimental test), it doesn't exist? This would seem to make headsonpikes the standard by which reality must be appraised and judged -- the old sophistical notion of "man is the measure" redux!!!

Is this what you mean to say? Thanks for writing, headsonpikes.

[betty boop]

So then there is such a thing as "doughnutness" by which we may only view a doughnut in the material world as a "contingent" example made possible by its "necessary" objective reality?

LOL StJacques! Methinks this is the case!

[betty boop]

The dichotomy [Plato v. Aristotle] is an existential one, not an ontological or even an epistemological one.

Well and truly said, Taliesan! Thanks for writing.

[StJacques]

". . . Sort of like discourse about Platonic 'forms'.... There's no 'there' there. . . ."

Well if the only criteria is that we can reach out and touch it, ok, point taken. It is difficult to affirm ideational reality. But I think it is even more to difficult to deny it, because if you do so, you must deny that mathematics works because it represents "objective truth" and is rather something that just functions as it does because of the way we define mathematical terms and construct systems of reasoning and you must also deny that Morality and Ethics work because they represent something intrinsic, i.e. "innate," that goes beyond physical existence. This last point is very important in my opinion because I have never been able to view a complete, coherent, and cohesive treatement of the "Problem of Evil" that I find satisfactory without viewing it through the prism of innate ideas.

These are two of the biggest reasons why I think Plato got it right in his Theory of Forms.

[cornelis]

Thanks for your last reply, StJacques. I notice that the subtle nuance between the material and immaterial world is often a convenient refuge whenever in a pinch. And the attempt to preserve for science a privileged domain or monopoly on "fact" and "cause" is no different that the presumption of truth claimed by enlightened theological dogmatists.

[cornelis]

Wow, bb, there's just so much here. This is truly eclectic and fecund in the same way. Such interesting subjects. I wish our collective character would turn to greater patience to distinguish between the possible positions staked out.

If--to better understand--I were to clarify anythying I would start with the term monism, (which incidentally surfaced in one of the replies above). You say, in the beginning that you make,"Thus does Plato (d. 347 B.C.) succinctly describe how all that exists is ultimately a single, living organism."

I would like to juxtapose the Aristotelian quibble about "separable forms" in the Metaphysics and Nicomachean Ethics. Those familiar with the Aristotle's writings will remember that Aristotle wanted to part ways on this aspect of separable forms. But in your description, which may or may not be more in line with later Platonism and Plotinian henism, do the concepts of a "beyond" and "trancendence" suggest some sort of dualism or can we somehow speak of both?

In short, I'm seeking a clarification on "all that exists" and a further determination of the scope of "single" in that first statement.

I'd love to know what you think.

[cornelis]

A note about infinity. Infinities have different orders.

Worth repeating.

[Eastbound]

"Monism is what happens when Jesus is considered to be no longer equal to God the Father. That is he is considered a great teacher/philosopher/prophet and very very good. The Moslems think of Jesus as being a wise man/teacher/prophet/very very very good--but not God. They take the assertion that Jesus is God in the same way that the Jews of Jesus time took Jesus words--as blasphemy."

Reminds me of the moment Jesus asked his disciples who the people thought he was. Similar answers as you give here. But Peter knew who he was and was the second to give that testimony. I think John the Baptist was the first when Jesus came to fulfill John's prophecy at the baptism. So there were at least two early witnesses who received that revelation before the crucifixion.

Can it be said during these latter times (and even through the history of the church) that though there be many who believe that Jesus is the Son of God, they have not yet come to the knowledge that He is the Son of God (the manifestation of the Father), as they have not yet received that Revelation?

It seems like such a simple step but to believe first, and then the walk of faith towards that Revelation. But those who think they are well have no need of a doctor and will continue holding to that which they are familiar with.

Yes, ckilmer, you've described many who still cling to the narrow view, and sadly enough, actually reject the promptings of the Holy Spirit which will lead them to the very threshhold.

[ckilmer]

Can it be said during these latter times (and even through the history of the church) that though there be many who believe that Jesus is the Son of God, they have not yet come to the knowledge that He is the Son of God (the manifestation of the Father), as they have not yet received that Revelation?
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well there is tremendous slight of hand by the liberals as to the meaning of "Jesus is the Son of God" so as to construe it to mean that Jesus is not coequal in the Godhead with God the Father. While this slight of hand happened in the seminaries of all the mainline protestant--I am most familiar with the Presbyterians. A good guy to give the skinny on how and when this came about is a guy by the name of CJ. Gresham Machen who wrote the book "Christianity and Liberalism"

That said, we are saved by faith alone (sola fida)

[Eastbound]

"That said, we are saved by faith alone (sola fida)"

Yes, Faith. Some say faith without works is dead. Others say I'll show you my faith by my works. Could it be they are both saying the same thing, if we can for a moment consider the possibility that faith and works are synonymous? By works I refer to the activity that one engages in, mental, spiritual, physical, (no doubt all three) that propels one toward the KNOWLEDGE and prepares one to receive that knowledge.

Upon receiving the knowledge, we find our belief was true, and faith (the works one does, the 'labors' investment) was justified.

For isn't our goal to RECEIVE the personal knowledge of the Author and Finisher of our faith in the same qualitative way as Peter, John the Baptist, Paul, the rest of the apostles, and the post-resurrection church?

Analogy: On payday we receive what we've been working for all week, believing our employer will render to us the fruits of our labors. Once receiving our paycheck, yes, we can say our labor investment (faith) was justified for now we KNOW the promise to pay was a true promise for we are now the possessor of (we have the KNOWLEDGE of) our reward.

So, obviously, if we need rent money, we must go to work to earn it. If we are sick, we must go to the doctor to be made whole. Faith could be another word meaning 'walking the proper path' to get to the doctor's house. For some of us, it's realy hard labor. (Heh!)

Thanks for the FAITH word, ckilmer. Looking at it from this angle helps, and may fill up a few pot holes in the path. And thanks again for your informative reply.

And a post script: Grace? Yes, God is gracious. Thus far we've had the time dispensation (another aspect of Grace) to become fully pursuaded in our own minds that the knowledge of Jesus Christ and He who sent Him is attainable while we are still here.

And further, Grace is the gift of God who stopped time (in a sense) perhaps in the same way time stops temporarily when we are in the 'grace period' when we stop paying our insurance premiums. The policy continues in force anyway -- until that dispensation ends.

[ckilmer]

it looks to me like you are talking about the process of sanctification.

a statue cannot carve itself. The master craftsman has to do the job.

Just so we cannot save ourselves. The best we can do is bring ourselves into the presence of the Lord and have him do the work on us. Our part, and it is no easy one--is to get into the presence of the Lord. It is Jesus who enables even the greatest of sinners to get into the presence of the Lord. This is the reason that Jesus is the great and gracious gift from the Lord. The hard truth here is that we can only love God because he first loved us. The calvinist catechism in this respect is we are saved "by grace (of God) through faith (in christ)"