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.

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About Citizen Tom

I am just an average citizen interested in promoting informed participation in the political process.
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26 Responses to WHAT IS MATHEMATICAL PROOF? DOES 2 + 2 = 4?

  1. zen says:

    So, is your position that the universe is an abstraction when defined by mathematics, yet a reality when described by the bible?

  2. Citizen Tom says:

    zen – The Bible is the inspired Word of God, but even the words of the Bible do not convey the reality of Christ, Our Savior. What the Bible does do is tell the story of redemption. That is a story each of us can make part of our reality.

  3. zen says:

    Okay, if that is what you believe.
    What I’m asking about is the physical world. Your post speaks of describing the physical world as an “abstraction.” And yet, you mention religion as if it were able to explain, or describe, more accurately than mathematics, the physical world.
    Just wondering if I follow you correctly.

  4. Karl Popper’s “Conjectures and Refutations” is a great book on how we know what we know – epistemology on ontology. We used it as a text when I taught ‘Research and Methodology for the Social Sciences’.

    Underlying the basic concepts of rational empiricism – given to us be the Greeks – is the Judeo-Christian concept of a created and ordered natural world – and all sciences.

    So, the quest for knowledge based on science and the scientific method, not magic, only happened in the civilizations that inherited the ideas of the Greek and Judeo-Christian civilizations. Which means only the West and the Islamic Civilization progressed – until the Muslim imams decided in 1250 that there was no new knowledge to gain and they have stagnated since then.

    The Reformation and American/British Enlightenment gave new energy to Christians seeking to discover what God had created in all sciences.

  5. kgotthardt says:

    “Only in the abstract does 2 + 2 = 4.”

    I LOVE IT!

    I recall having an argument in high school that if we believed 2+2 were 5 and not 4, then we (as humans) would accept it as fact because math is a human made concept. The answer I got from my brother and Catholic school teachers was, “You need to take it on faith.” Now I find THAT an interesting equation.

  6. Citizen Tom says:

    zen – I think James answered your question.

  7. “The belief that the universe is orderly and follows rules actually began as religious belief.”

    Actually, no. The converse, in fact; religious belief began as a desire for an orderly universe. Mankind wanted dawn to have a reason and purpose, and so it invented Ra. Mankind wanted a deity, and so it came up with God.

    As for this 2 +2 = 4 thing, I don’t think your argument there is sound, either. I get what people mean when they say “But if we believed 2 +2 = 5 and not 4, then we would accept that as fact because math is a human made [sic] concept,” but not exactly. I also get what you’re saying about abstract principles, and 2 miles + 2 miles equaling 4 miles because they’re not, like, Platonic miles, but not there, either. The first argument (2+2=5) mistakes symbolism for abstraction. I can literally hand someone 2 apples and then hand them 2 more, and they would have 4. You could call that whatever you’d like (3, red, chair), but still in the real world, it’s representative of the amount of apples.

    The second argument presupposes miles are imperfect, which is fine, but math is not.

  8. kgotthardt says:

    But symbolism is also mankind-made. Since we filter all symbols and ideas through our brains and perceptions, math can be nothing other than just abstract symbols we have created and given meaning to.

    Here is one definition: “A numeral system (or system of numeration) is a mathematical notation for representing numbers of a given set by symbols in a consistent manner.” (Wikipedia) So math is a system of symbols, the difference being, it is consistent; but the consistency is something humankind has developed and someone agreed upon. Like language, there is an accepted system. Note advanced thinkers like Einstein tipped accepted theory on its head and thus came up with new ideas about math, science, and physics. Through poetry, writers do this all the time.

    “Piaget suggested that infants are born with no understanding of numerosity, which is the ability to discriminate arrays of objects on the basis of the quantity of items presented�for example, being aware of that a quantity of two is different than a quantity of three. Early Piaget experiments (Piaget 1942) described infants’ lack of numerosity as a poor perception of quantity conservation.” http://web.media.mit.edu/~stefanm/society/som_final.html In essence, because infants have no language, they cannot think in the symbols we attribute to quantity. They are TAUGHT to believe numbers have words and patterns associated with quantity. While infants can sometimes distinguish between more or less, numeracy must be learned. Put any older child or illiterate human in the forest and s/he will be able identify which herd of deer is larger or smaller, but s/he won’t be able to count or communicate the numbers. Counting is learned behavior similar to the development of phonological awareness.

  9. kgotthardt says:

    One more interesting quote from the MIT article: “The numeric systems invented vary across time and place, and there is no doubt that the properties of such a system can facilitate or impede the development of children�s mathematical understanding.” So math, no matter how consistent, does indeed change. It’s not as perfect or pre-determined as we might like to believe.

  10. Citizen Tom says:

    Will – You raised two interesting points.

    First, you raised the possibility that god is man’s invention. Before you accept that proposition, I ask that you consider how ancient men worshipped their gods. Have not men always feared the unknown? Don’t we wanted certainty? Don’t we want power over our lives — and over others? So did the ancients. So they invented gods they could bribe. For such gods and the power they believed they offer, some of the ancients would sacrifice anything, including their children. Such gods did not order the universe; they created chaos.

    God, however, is not a god who offers us power. Instead, He is a God who loves us and wants us to love Him in return. He offers us the opportunity to learn how to serve. We cannot bribe God anymore than a child can bribe a good parent. For our benefit, He created a universe, an orderly universe, where we can learn how to love.

    Your second point I think kgotthardt addressed quite adequately. So I will just ask you about apples. Unless we are talking about apples in the abstract, are all apples equal? Don’t some apples taste better? And if not all apples are equal, how can you say that two apples plus two apples exactly equal four apples?

  11. kgotthardt says:

    Tom, what do you think of this idea: “God is the term we use to explain the inexplicable.”

  12. Citizen Tom says:

    “God is the term we use to explain the inexplicable.”

    kgotthardt – I think I know where you are coming from, and perhaps that statement is partly true. However, for atheists, I fear that statement is the beginning and the end of God. For an atheist, God is just a dodge for being unable to explain the inexplicable.

    While much is unexplained and ever so puzzling, God has told the important things. He has told us He exists. He has told us why He created us. He has told us how He redeemed us. And He told us what we must do.

  13. Just talking about apples. Not apples in the abstract, and neither size nor taste have anything to do with it. Take 2 apples out of your refrigerator. Put them on the counter. Now take 2 more out of the refrigerator and put them on the counter, too. And now you’ll have 4 apples on your counter. Or chair apples, if you prefer.

    And sure, maybe infants have to learn numbers and math, but, well, I’m not an infant. So.

    And actually, I didn’t introduce the invention of God; I merely extrapolated what you were saying. You said: “The belief that the universe is orderly and follows rules actually began as religious belief.” I was just pointing out that you were putting the cart before the horse, because, in fact, religious belief began as the desire for orders and rules, and not vice-versa (as the above statement implies). The Egyptians wanted rules for their universe, for their sunrise, so they invented Ra and etc. The Norse wanted rules for the afterlife and the end of the world, so they invented Valhalla and Fenrir. The Native Americans wanted an explanation for how the world began, so they came up with the turtle with the world on its back; the Hebrews, when faced with the same task, came up with a paternal father figure who said “Let there be Light.” And when certain people believed they required salvation they couldn’t earn on their own, they ascribed the mythological stories of Baldur and Dionysus to a humble carpenter from a Jewish city under Roman rule.

    Faith is a story we tell ourselves to explain the world.

    What’s unexplained or puzzling? When I hear people say anything is inexplicable, I tend to wonder if they know enough to explain it.

    Really. School helps. Education is awesome.

  14. kgotthardt says:

    Tom, a recent study (albeit somewhat limited) published in the Washington Post showed atheists largely believe in a higher power. They might not call it “God” but there is something there. Agnostics generally don’t know what they believe and are comfortable with that. And in my mind, anyone who has taken the time to think about the existence of a God probably is somewhat concerned about the meaning of life and what it should mean. I try not to get hung up on anyone’s journey towards truth because it’s highly personalized. So long as they don’t impede on anyone else’s journey and try to do right in life, I have no problem with them.

    Will, I am not sure what you mean by education. I’ve studied world religions and history. I’ve done some time in Catholic School and “taste tested” a number of Christian churches as well. I don’t think education necessarily leads one away from the idea of God. In fact, education can strengthen belief because education challenges people to think for themselves and sort out, “Is this really what I believe?” Most teens and people in their 20’s go through this process with or without organized religion and education, so we might as well encourage thought and inquiry. As you can see, there is plenty of it going on in this blog.

    Religious history and thought are fascinating aspects of our culture and I wouldn’t want to do without. That said, limiting ourselves to learning one perspective is just that–limiting ourselves.

  15. kgotthardt says:

    BTW, Will, I will assume you must have been an infant at ONE time or another. Since you are apparently concerned with origins, I am surprised you dismiss your own so easily.

  16. Citizen Tom says:

    Will – Is art done for its own sake? Do people do math just for the sake of doing math? Well, I suppose a few do, but most of us do math with a purpose in mind. If you do not consider the nature of the apples you count worthy of your notice, what is the point of counting them? Would you pay the same price for four rotten apples that you would pay for four sweet and juicy apples?

    Religion such as you describe has little to do with an orderly universe. Perhaps that has to do with your understanding of “faith.”

    Faith is word that means different things in different contexts. In one context, faith is a virtue. Consider what it means to have faith.

    Imagine a difficult moment. You have trained for a difficult task, and you are about to be tested. Now, to overcome your fears and concentrate on the task at hand, you must have faith in yourself. You know you can do what is necessary, but you must still overcome your doubts. You must have faith in yourself.

    We are each fragile. The universe is beyond our knowing. What good does it do to have faith in a mere man or woman, that face that slowly ages before us each morning in the mirror? When know our failure is altogether certain, what cause is there to have faith? So we strive for something greater than ourselves.

    Christians say God known to them. So they describe faith in God as trusting in what we know to be true. Such faith allows Christian to concentrate on the task at hand, to live the life that God would have us live, in the service of others.

    kgotthardt – Although Jesus said we must each make our own choice to accept salvation, he also gave us the Great Commission. So there is a balance. We cannot be indifferent. So what is the correct balance? Well, my guess that it is pointless to try to argue or pester anyone into heaven, and I suppose that is what you meant by not getting too hung up.

  17. kgotthardt says:

    “Well, my guess that it is pointless to try to argue or pester anyone into heaven..”

    Yes, but also, you can’t MAKE someone believe in a specific heaven or hell. My mother has tried for years. It doesn’t work : )

    Would you try to make a Hindu believe in your version of heaven or hell, for example? Sure, you could talk about ideas from your belief system (and I personally might welcome it), but the idea of what I call “persuasive conversion” has never appealed to me. In other words, “Don’t badger me!” Does that make sense? Not everyone is Christian, after all. And I don’t think God INTENDED everyone to be Christian.

  18. Citizen Tom says:

    kgotthardt – One thing I hope I have learned from reading the Bible is a bit of humility. When you suggest that we might know what God intended, I think of the Book of Job and laugh.

    What did God intend? I would imagine that what is is what God intended, but I do not know why He did things the way He did them.

  19. kgotthardt says:

    LOL! Great point!!!!!

  20. “Will, I am not sure what you mean by education. I’ve studied world religions and history.”

    Science helps, too.

    The problem, Tom, is that you don’t know anything to be true. That’s the point. And four apples is four apples regardless of their state of decay. I wouldn’t buy four rotten apples to consume, but that’s a completely different issue.

  21. Citizen Tom says:

    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

  22. kgotthardt says:

    “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 : )

  23. 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.

  24. kgotthardt says:

    “I pursue Christendom.”

    Amen to that, Will. That will save the world.

  25. davidullery says:

    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.

    • Citizen Tom says:

      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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