WHAT IS MATHEMATICAL PROOF? DOES 2 + 2 = 4?

cross.pngWhen I investigated the Math Investigations program being implemented by the Prince William County Public Schools (PWCS), I discovered Contructivism as the underlying basis for this program. Constructivists believe we learn by building upon the knowledge that results from previous experience. I found this idea intriguing. As I studied the ideas of the Constructivists, however, I notice a gaping hole in how they put their ideas into practice. What role does religion have as a foundation for learning?

Religion and Logic

Consider the issue of proof. In this article (here), two of the academics who initiated Investigations in Number, Data, and Space® (PWCS’ current textbook series for elementary school children), Michael T. Battista and Douglas H. Clements, discuss how to teach children the concept of mathematical proof. In this article, they approach the issue at a rather high level. For example, they assume children will accept the notion that something can be logically proved.

The belief that the universe is orderly and follows rules actually began as religious belief. Our ancestors originally believed in a universe ruled by numerous gods and spirits (Roman Mythology, for example). With such beliefs they found any odd happenstance easily explained. “The gods did it.” Nature had no rules, just fickle gods and spirits. The belief that there is only one God and that this God loves us provided a new perspective. It made sense to apply Aristotelian logic. In fact, some people even went so far as to assert that the existance of orderly universe proves the existence of God (see here).

The Validity of Mathematical Proof

It might also be interesting to hear Battista and Clements discuss the validity of mathematical proof. Mathematics is very much like a language. Consider the first task God set for Adam (Genesis 2:19-20). Adam named the creatures God had created, and with this task God set man above all the other animals.

When we think rationally, we think in words. Each noun provides us a symbol denoting a model of something we find in the real world. Each verb provides a symbol that models a process we can find in the real world. Adjectives and adverbs narrow the meaning of the words they “modify.” If I say “tall building,” then the adjective “tall” helps the listener better understand something about a particular building. Similarly, if I say a cup quickly overflowed, my listeners know more about how fast the water was running.

Like any other language, mathematics provides us symbols and rules that help us to model the real world in greater detail and with greater accuracy. Also like any other language, mathematical symbols are only abstractions. Only in the abstract does 2 + 2 = 4.

When we take a measurement in the real world, that measurement is an approximation. We cannot exactly measure two pounds, two inches or two quarts. We can only measure abstractions with infinite accuracy. Thus, because money represents an abstract concept, two dollars is exactly two dollars. So while two dollars plus two dollars is four dollars, two miles plus two miles is only approximately four miles. Because the term “miles” models a real distance, our measurement must contain some degree of error. Because we use mathematics to model the real world, what we find in the real world, not with mathematics, sets the standard of proof.

Abstraction Versus Reality

What do children learn about proof? With television, computers and even books, we immerse our children in abstractions. In the abstract world of fiction, we teach children of an imaginary universe of perfect heroes, of talking animals, and of spotless worlds. In the real world, our children learn that is impossible for them to draw a perfectly straight line. In the real world, children learn that their teachers are flawed, animals bite, and the water they drink contains poison. As part of a flawed real world, do our children learn that their own imperfections contribute to the world’s imperfections? Are we providing our children proof that they are unworthy?

Properly taught, mathematics helps children learn to deal with the frustrations posed by abstractions. When we teach children how to relate mathematics to the real world, we help children to learn abstractions are not real. Abstractions are merely tools. We help children to understand that an abstraction serves no useful purpose unless it helps us to understand God’s creations. The only proof that matters is how well an abstraction models the universe God made for us.

26 thoughts on “WHAT IS MATHEMATICAL PROOF? DOES 2 + 2 = 4?

  1. If Adam has 2 apples and Even gives Adam 2 more apples, then Adam has exactly 4 apples. Mathematics maps into the real world for discrete objects exactly. Measuring distance at at the macroscopic level will produce error because measurements of that sort are non-descrete. That is one reason why Newton had to invent (co-invent) the Calculus.

    “When we take a measurement in the real world, that measurement is an approximation. We cannot exactly measure two pounds, two inches or two quarts. We can only measure abstractions with infinite accuracy. Thus, because money represents an abstract concept, two dollars is exactly two dollars. ”

    Notice how all of these examples are measurements of the non-descrete (analog) kind. The reason money can be measured exactly is not because it is abstract, it is because the units are descrete – just like the apple example that I give above.

    One can say the same thing for any book, including the Bible, since that book must be written down in a language and language is abstract. No two people will understand what they are reading in exactly the same way from each other or in the same way as the author(s).

    Actually, it is words like “tall” as in a tall building are fuzzy and non-precise. Is a man who is 6 feet in height tall? For the sake of argument, let us suppose he is. What about five foot eleven? 5′ 10″? If you say a man is 6 foot tall, then you know much more precisely the man’s height. We can do the same thing for measuring colors. If we say the apple is red we are not being as precise as we would be, if provide a range of light frequency and define that as red. In fact, if we define colors by wave frequency, then we can program a computer to distiguish colors to arbitrary precision. The same will apply for measuring the rate in which a container is overflowing. If we say “rapidly” we get some idea of what is meant, but if we give a nearly exact rate in say liters/second we can get a much more precise measurement. In principal, we can measure all non-descrete items down to their Planck units (i.e. the smallest possible length, time) and anything smaller than those units are meaningless (there is no information smaller than those units – very similar to the concept that you cannot make anything colder than absolute zero.) Ultimately the Universe is composed of descrete units of space, time and energy.

    It is interesting how Christian like to cherry pick the Bible. They will accept the 10 Commandments as true, but ignore the commanded punishments (e.g death by stoning). According to this article, there are not exactly 10 commandments, since are both abstract language and numbers involved. For this reason, and others, there are several different denominations. Different groups cherry pick the bits that are pleasant and either do not read the rest, choose to ignore what it says, or “hand wave” it away.

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    1. David – Thank you for visiting.

      On the matter of mathematics, I think we are largely in agreement. Perfect is nearly impossible. Consider this statement of your own. As you stated: “no two people will understand what they are reading in exactly the same way .”

      Where I think we differ is on how our abstractions affect our mathematics. The concept of a discrete unit is itself an abstraction. For a discrete units to exist in our minds, we must presume, for example, that all apples are equally tasty and nutritious, that none contain worms or rot. If a grocery store violates that assumption, its customers will complain.

      Unless we choose to coin gold and silver, to use a currency that has value in and of itself, then money is wholly an abstraction. So long as they are recognized as currency, we are indifferent as to the condition of paper currency and coins. We just try to replace our currency before it wears out.

      As you suggest, mathematics exists as a subset of our language. Language exists to communicate our thoughts. Because our thoughts begin as abstractions from reality, even with mathematics, we can communicate only so well.

      You end with a criticism of Christians that deserves further comment.

      It is interesting how Christian like to cherry pick the Bible. They will accept the 10 Commandments as true, but ignore the commanded punishments (e.g death by stoning). According to this article, there are not exactly 10 commandments, since are both abstract language and numbers involved. For this reason, and others, there are several different denominations. Different groups cherry pick the bits that are pleasant and either do not read the rest, choose to ignore what it says, or “hand wave” it away.

      As you noted yourself, we each interpret what we read differently. Because we each have our own unique vision and viewpoint, this is an inevitable fact of life. Because we are also weak, serious misinterpretations and cherry picking of the Bible are difficult to avoid.

      Are all the commandments equal? We think in quantities. We set priorities. Some commandments do, for example, actually seem to have more weight than others. Some commandments direct us to love God. Some commandments direct us to love each other. Nonetheless, God wants us to obey all His commandments — not just those we think important.

      Note that there is a profound difference between the Old and the New Testaments. When the Old Testament Hebrews tried to live by the Law, they found it impossible to live perfect lives. Yet each sin requires punishment — or a sacrifice to God for the remission of our sins. In recompense for their sins — to appease the Almighty — the Jews and others made innumerable sacrifices. Yet nothing they could do was enough. So they felt compelled to come to the alter and to sacrifice again and again.

      The New Testament established a new covenant. With the New Testament, we do not ignore the punishment for sins. We recognize that price has already been fully paid. Jesus paid the price.

      With Jesus’ death we see an illustration of our inability to model discrete quantities. Jesus offered up His Life only once, but His Sacrifice paid the full price.

      So what is left for us to do? Our task is to accept His Gift for the remission of our sins. Our command is to do as He said, to love God and each other.

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  2. I’ve already surrendered. I don’t believe in Christianity as proposed by those who claim salvation; I accept my place in Christendom.

    So many Christians seem to forget that “Christ” is a title, and not a person.

    I don’t oppose Christ, in the least. I pursue Christendom.

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  3. “Science helps, too.” Of course it does. It helps even more if you plan to become a scientist. I think Tom’s background as an engineer qualifies him, so I see the both of you as going round and round with this because it comes down to whether you hold religious beliefs (in this case, Christian) or not.

    Personally, I believe no one apple is like another, so you can add as much as you like but still not prove anything other than you can add : )

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  4. Will – Because we each battle against truth, we educate ourselves by surrendering to it.

    Because it does affect how and what we count, we separate the abstraction, math, from the real world. Nonetheless, in application the abstract must surrender to the real world, not the other way around.

    When buyers buy fruit, they first grade that fruit. If they buy in quantity, then they develop systematic ways to grade the fruit. Similarly, when you buy eggs, they are separated according size and quality. We all know a large egg is sort of like two small eggs. We also know people will pay more for brown eggs.

    Last Sunday, my pastor acquainted me with a quote from a man I had never heard of before. Based upon the Wikipedia article on him, he seems to have been something of a character. In fact, he seems to have been rather skeptical of the idea we might know anything.

    Here is what he had to say about proof.

    “All truth passes through three stages. First, it is ridiculed. Second, it is violently opposed. Third, it is accepted as being self-evident.” — Arthur Schopenhauer

    Since you regard yourself as a writer, I presume you are also a reader. I expect you have read the Bible. Are you familar with the story of Paul’s moment of conversion? Prior to this moment, Saul violently opposed Christ. After that moment, Paul had set aside his pride and surrendered to Christ. Will you ever experience your moment of surrender. I don’t know; the matter is wholly up to you. No one can make us accept a truth.

    “A man convinced against his will
    is of the same opinion still.” — Unknown

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