Beginning Mathematics/Where To Start? With Nothing!
Where To Start? With Nothing!
[edit | edit source]Clear your mind and assume nothing. When you achieve this blank state you have arrived at the starting point for Mathematics: the big Nothing (with a capital "N"). To move on from Nothing using the Language of Mathematics we will need a framework to build on. The framework we need has to allow us to expand the reach of the language as we go. To find such a framework Mathematics looks to the world of Logic.
A Brief Glimpse of Logic
[edit | edit source]First, a description of Logic1:
[Logic] is most often said to be the study of arguments, although the exact definition of logic is a matter of controversy among philosophers. [The task] of the logician is the same: to advance an account of valid and fallacious inference to allow one to distinguish good from bad arguments.
From this description it is clear that a solid system of logic requires arguments to work with. A Logical Argument is the ordered process of going from some definitions to reaching a conclusion. Obviously you can't have an argument (or reach a conclusion!) without something to argue over. A Definition is a simple statement and can form the basis for an argument. A Conclusion is the endpoint of an argument. A useful argument results in a conclusion which tells us something about the original definitions.
An Example is in order:
- Definition: Something which has water on it is wet.
- Definition: Something not protected from the rain gets water on it.
- Conclusion: Something not protected from the rain gets wet.
These three statements make up a logical argument. Using the conclusion from this logical argument, we can now identify the things around us in jeopardy of getting wet the next time it rains. The couch in the house is safe; the trees outside are not. Notice that the conclusion comes directly from the definitions -- for this example there is no Body to the Argument.
Thinking deeper about this argument it becomes clear that we jumped into the middle of something. You could ask: What is Rain? What is Water? What is a "Something"?
Obviously there are a lot of logical arguments which could have preceded this one. For example, we could define Water as a collection of molecules with the chemical formula of H2O and define rain as a collection of collections of H2O molecules. An argument relating rain to water can then be made.
But then what is a Molecule? Again, we could define a Molecule as a collection of atoms. Still, what is an Atom? For that matter, what is a Collection?? It quickly becomes obvious why philosophers spend so much time quietly thinking to themselves.
Fortunately we're thinking about Mathematics, not the Meaning of Existence (which can easily take a lifetime2). Remember that a useful logical argument results in a conclusion which tells us something about the original definitions. In our example above we discovered that we can combine the two definitions into a more compact form as the conclusion. In this case, it's hard to say that the conclusion is much more useful than the definitions. However, some logical arguments can take many, many definitions and arrive at a simple conclusion which itself becomes tremendously useful. This is called Reduction3. Reduction will be the first goal of our Mathematics framework.
Starting The Framework For Mathematics
[edit | edit source]Time to start building! Since the starting point is Nothing there can be no arguments. We need a definition!
Definition: An Object is something that can be described.
Any logical argument leading to this (as a conclusion) we will disregard as being in the realm of pure Philosophy. We will also assert (not assume!) this definition is true. Consider the previous assertion of truth carefully! The implication is powerful: from the very beginning Mathematics ceases to be tangible. With this definition we can say "The Universe is an Object" or "Emptiness is an Object" or "Words are Objects"; indeed, even a Thought, a Color, a Feeling, or even Something That Has Not Been Discovered is an object! Ahhh, the freedom! No boundaries! If this fundamental definition is preserved (and it will be) our Mathematical Framework will have tremendous reach and breadth.
Wait! There is one problem -- our definition is written in words, specifically English, and we don't want to be limited as to who can understand or use this definition. There are concepts in other languages as well as things undiscovered which we want to include in the realm of Mathematics. To keep the limitation of language from putting boundaries on the framework, symbols are used in place of words. For now, we will use lowercase English letters to represent objects. Remember, they are symbols in Mathematics, not letters. Letters have sound; symbols do not. Symbols can represent any Object as defined above. We can now add a definition to our framework:
Definition: A Symbol is a written character.
Next comes a decision on how to use Symbols which brings up another problem: there are not nearly (not even close) to enough symbols to give every Object its own symbol. Adding to the problem is the confusion caused in other people if symbols are assigned to objects. Other people will have to know what each symbol specifically represents, which means creating a gigantic Symbol Dictionary. The Symbol Dictionary would have to be updated every time a new object is found, created, or considered. That is no good!
To solve the problem of Symbols, we add the following definition to our framework:
Definition: A Variable is a Symbol which represents a property of some object(s).
Whew!! Our Dictionary just got much, much smaller. Reality requires us to have at least some symbols that are specifically defined and universal -- those situations are not far away. However, any symbol that does not have such a concrete definition can be used as a variable. The key point of the definition of a variable is "represents a property". Fortunately this definition is wide open since "represents a property" is not a restrictive phrase. Success!! Our language is universal AND our freedom has returned.
Putting the previous definitions to use we can now define a variable:
Let β (the Greek letter Beta) represent an object with two and only two wheels.
This statement does not require us to state exactly what object with two and only two wheels is represented by β. In fact, an object satisfying β does not even have to exist. But now we can tell what objects β could be. For example, β could be a bicycle, but not a car. β could be a motorcycle but not a motorboat. Notice the definition of β allows us to separate all objects into two groups of objects: one group which satisfies the conditions of β and another group which does not. This leads us to our next definition:
Definition: A Set is a group of objects with one or more common properties.
Sets are vitally important in Mathematics. If you are not confident in how this definition was reached it is probably well worth your time to review some of the previous definitions for clarification! Sets are so important and occur so frequently in Mathematics they require universal symbols to represent them. For a Set, the universal symbols include the left and right curly braces used as follows:
Definition: The Set B contains all objects represented by variable β.
(Almost) Mathematical Notation: B is the same as {β}
The Mathematical Notation is not pure -- it is "Almost" because it contains English words. Let's fix that immediately!
Definition: Equivalence means two symbols represent the same value, state, or are the same thing. Another common term for this condition is Equal. In Mathematics, the symbol = represents Exact Equivalence. Let's look at this more closely. Since "equivalence" can refer to the "state or condition" two boxes containing three items in each could be considered "equivalent". However, if the three items are not the same kinds of things, say three cats in one and three dogs in the other, they are not "equal". For them to be equal, they must be exactly the same. This concept is recurrent in more advanced mathematical structures. This distinction is made now to avoid future confusion the student might have. Many students get the impression that mathematics in not consistent and contradictory when they have such an experience. Keep in mind that people sometimes take shortcuts when explaining material or are so caught up in the explanation at hand that things slip their mind. This is human nature and should be forgiven since perfection seldom exists.
In our (Almost) Mathematical Notation for Set B above, the phrase "is the same as" shows equivalence. Therefore, we can write the condition in pure Mathematical Language:
B = {β}
Hopefully your mind hears "B is a Set of all objects that satisfy the conditions of the variable β". It is common in Mathematical literature to use uppercase English letters as symbols for Sets, and lowercase English letters as symbols for variables. Also, it is common for the Set symbol to be the uppercase equivalent of the Variable symbol. We will follow these conventions from this point forward.
Example:
A = {a}
How did your mind hear that line? It is significant to note that Variable a has never been defined, yet we understand the concept of what is represented by the line above. It is important to understand that the line above also has not validated any of our previous definitions. This is a small taste of a special power of Mathematics:
Definition: Generalization is the ability of Mathematics to specify behaviors and relationships for a broad variety of objects without actually defining what the objects are.
There are not enough tools in our framework at this time to exploit Generalization completely. But we will eventually! For now, here's the definition of another set:
P = {p}
This is an abstract Set, since we do not know what the Variable p represents.
Q = {q:
all wordsany word that starts with 'm'}
This is a concrete Set since a definition of Variable q is given. A new universal symbol has been presented here.
Definition: In the notation for a Set, a colon (:) is used to represent the phrase "is defined as".
Using this definition, the Mathematical Language given above for Set Q reads "Q is a Set of all objects that satisfy the conditions of the variable q which is defined as all words that start any word that starts with 'm'".
There is an additional way to specify a Set when there are a specific group of objects contained within it.
X = {x: a donkey, a mushroom, a glass of water, a cloud}
There is NO spoken phrase for Variable x that correctly describes these objects in terms of a common or shared property. Nonetheless, this is a valid Mathematical Set. Again this demonstrates the ability of Mathematics to express concepts that formal written language cannot describe. This also demonstrates a small piece of the power of Mathematics which we will put to great use as we move forward.
References =
[edit | edit source]If the sources have changed, some quotes may not be word for word; please be forgiving!
- 1See Wikipedia:Logic (Return)
For college level math, it is necessary to become somewhat familiar with Greek as well as Latin letters (which is what the English alphabet is derived from), as both are used as Mathematical symbols. When learning new concepts, anxieties can be reduced if the symbols, at least, are familiar.
Wikipedia:Greek_Alphabet (Return)