Tuesday, December 21, 2010

Introducing Mathematical Fallacy: The Logical Oversight of Scientific Deduction

Within everything accepted, you'll find everything that is overlooked.

Part of the problem in fallacious redundancy is that it’s often well hidden within the logic of the argument. An argument’s basic logical composition even in its simplest form can be rather misleading. A correct logical conclusion can be evaluated as such, (is, is, therefore). Consider the arguments make up as follows, (x is, y is, therefore x is y is). Let’s say we describe the conclusion (x is y is) as z. We could then describe the argument as follows (x is, y is, therefore z is). That’s simple enough to understand, but that framework of thought has its disadvantages, primarily, the ability for one to define a rational argument and properly define “z” is dependent on 2 things, x and y being absolutely correct. If either x or y have not been fully evaluated logically, their argument can no longer be consider a rational constant, a rational constant is an expression that is predisposed and defined under parameters that support all of its supporting conditions as true. A rational constant comes from the proper evaluation of the form we just went over (x is, y is, therefore z). If ‘x’ and ‘y’ are rational constants, then we can determine ‘z’ subsequently is therefore a rational constant as well. Defining a rational constant is extremely difficult, because in order to do so, you must unquestionably understand all of its influences perfectly. I submit that nothing is more difficult than a fully evaluated logical thought. The irony of simplicity is nothing is more complex.

Can anyone claim to understand anything in perfect terms? I admit, that’s a rather unfair question, as I do not, nor do I believe anyone in fact possesses the ability to discredit someone who would say otherwise. This is why in my opinion the importance of asking a logical question, greatly outweighs the importance of any suggested logical conclusion. As I do not believe anyone possesses the ability to judge any logical conclusion fully, the ability to ask a logical question is never limited to the individual, as it does not predispose the necessity of a logical conclusion to support it's asking. The logical question has its own drawbacks, and its consequences cannot be overlooked. The logical question will not create a rational constant as it has no means of doing so. The ability to answer the logical question only refines the ability to ask the next logical question. The problem arises in its axiom as it does not disprove what is currently considered logical. Instead the logical question opens the conclusion to more questioning, and if asked well, the logical question provides further insight into the previous conclusions fallacy.

In its strictest terms, the logical question is not, nor should it be, considered scientific, as it does not hold to the scientific process. The logical question does not draw new knowledge rather if done correct, it highlights where there might be misunderstanding. While it probably isn’t difficult to see how this might be helpful, the logical question more appropriately is defined as a check in scientific understanding in the pursuit of balancing scientific knowledge. I submit that this type of evaluation is currently lacking in our modern school of thought as it is often neglected in our rational framework in nearly every respective field, philosophy notwithstanding. Philosophy though does not provide further scientific insight in a quantitative way, as is required in any scientific pursuit, or more importantly scientific conclusion. Early philosophers such as Socrates created understanding in these terms, but as the field of science developed, this type of questioning has become ever more neglected as it does not lend itself easily to the scientific process. I submit that the lack of development in pursuing the logical question has created gaps not only in our understanding, but the effectiveness in questioning our understanding. We will look at this effect in our modern understanding of science moving forward, but we must first ask a very important question. If in fact the scientific process allows fallacious arguments to support what appears to be a rational conclusion as we just discussed, where in the process are these fallacious arguments introduced? For your consideration, I submit the fallacy of the mathematical argument.

I imagine that at some point in your life, a math teacher showed you the mathematical example that seemed to prove 2=1. In case you haven’t seen it, here it is below.

Proof that 2 = 1

a = b

a^2 = a*b

a^2-b^2 = a*b-b^2

(a+b)(a-b) = b(a-b)

(a+b) = b

a+a = a

2a = a

2 = 1

Initially it looks almost like it might be correct. In my class we were given extra credit if we could show where the mathematical rule was broken. If you don’t know where this example is “wrong,” you’ll see in a moment. For our consideration let’s consider math itself. Math is the language of science. Math is a borderless language, and is used in every one of our systems. Math is the logic of the universe, as math does not lend itself to influence unless influence is introduced. What better language to evaluate science? Few if any would argue one exists. Let’s go back to our example and see why by mathematical rules, we assert 2 ≠ 1.

a = b

multiply both sides by a

a^2 = a*b

subtract b^2 from both sides

a^2-b^2 = a*b-b^2

apply the distributive law to both sides

(a+b)(a-b) = b(a-b)

divide both sides by (a-b)

(a+b) = b

substitute all a's for b's (remember, if a = b you can do this)/

a+a = a

regroup the two a's in the left side, and rename it 2a

2a = a

divide both sides by a

2 = 1

Take note of the step that says, “divide both sides by (a-b).” That’s where “the rule was broken.” The rationale is simple, you aren’t allowed to divide by zero, so the mathematical proof doesn’t follow the rules, therefore is incorrect. Now let me ask an important question, can you rationally disprove that 2 does not equal 1? Does 2 DNE 1 make logical sense? The answer might surprise you. At the bottom of this entry you’ll find what are referred to as ‘truth operators.’ They are to logic what (+, -, x, /) is to math. Below that you’ll find what is called a “Truth Table.” A truth table is simply these operators expressed in their logical form. Let’s look at this same expression in logical terms using math at the evaluator. Please refer to the truth operators at the bottom if at any point the steps become unclear. We will use Ta for true expressions of A and Tb for true expressions of B, naturally we will also use and Fa for false expressions of A and Fb for false expressions of B.

a = b

Ta=Tb

multiply both sides by a

a^2 = a*b

Ta^2=Tb*Ta

subtract b^2 from both sides

a^2-b^2 = a*b-b^2

Ta^2 – (Tb^2) = Ta*Tb – (Tb^2)

*apply the distributive law to both sides*

(a+b)(a-b) = b(a-b)

We must stop here though, because we’ve broken a rule of logic by applying the distributive property to both sides. The mathematical rule would suggest that the logical argument is as shown below.

(Ta + Tb)*(Ta – Tb) = Tb(Ta – TB)

This is incorrect though, by applying the distributive law to both sides, the mathematical rule asserts that it is appropriate to apply a false argument to a true argument and that application will result in a true argument. Do you see where it does so? See below.

Ta^2 – (Tb^2) = Ta*Tb – (Tb^2) is said to result in (Ta + Tb)*(Ta – Tb) = Tb(Ta – Tb)

The correct structure logically would appear as seen below:

(Ta + Fb)*(Ta – Fb) = Tb(Ta – Fb)

If evaluated properly evaluated the argument would then result:

Ta^2 – TaFb + FbTa – Fb^2 = TbTa – Fb^2

Which then cancels out logically to become:

Ta^2 – Fb^2 = TbTa – Fb^2

-Fb^2 cancels out to leave:

Ta^2 = TbTa

Which is the same as:

Ta*Ta=Tb*Ta

Cancelling out or rather, simplifying respective terms:

Ta=Tb

As you can see the logical deduction leaves us with the original argument.

Why have we reconsidered the steps as evaluated? We’ve done so because to apply     -(Tb^2) using the distributive law introduces to each term a false argument. Applying a false argument to a true argument (using the distributive rule) introduces characteristics that can be described using almost any logical rule, (non sequitur, amphiboly, etc.) Specifically though in terms of the mathematical application this arguably can best be described as syllogistic reasoning, wherein the rationalization of the rule is predetermined by what is deemed the correct answer, and subsequently the rules that follow are then used to support that argument. This is evident in what follows the introduction of the false argument in the next step. The false argument has been defined, and now what follows is the process in which it is removed. The next step begins this process, and is falsely attributed to where the initial argument of A=B, or where 1=2 is wrong. The next step is below.
divide both sides by (a-b)

(a+b) = b

As you see, it is then concluded that 1 DNE (Does Not Equal) 2, cannot be correct if using mathematical rule as the framework for the problem. The rule asserts that the proof violates mathematical law in that it is incorrect to divide by zero. Consider though, the framework of the original problem. A=B does not allow for this conclusion to be logically valid. While mathematically it is correct to allow substitution of ‘a’ for ‘b’ or vice versa, logically it is incorrect to suppose that the value of a or b might be substituted. By using the same reasoning that mathematically asserts it is incorrect to divide by (a-b), a new variable has been introduced, as b might be substituted for a, resulting in (a-a) or (1-1). This redefines the argument of a=b. It does so by not recognizing the definition assigned to the variables of ‘a’ and ‘b.’ It might be correct to substitute 'a' for 'b' for mathematical variables when a=b, but the initial proof does not require variables, 'a' and 'b' are not introduced until it becomes necessary to disprove what the constants assert. While it is my hope that by highlighting this, the fallacious reasoning becomes readily apparent, on the chance that it doesn't, let's look at what the mathematical proof that 2 does not equal 1, expects us to allow. By redefining 2 and 1 as constants into variables, you strip the definition of their meaning. By doing so you are essentially saying instead of evaluating this arguments constants, let's look at the argument as if it isn't defined at all. If you're willing to do this, you need not even evaluate the problem as you've already defined the answer. If you can say 2 = a and 1 = b, and then say a=b, why take it any further? Why not simply say, well if a=2 and a=b, then b=2, so a=b. It is proper to create rules where all variables are applcable to all constants, but to infer an argument of constant deduction under the guise of variable rule is not only redundant, but ignores the mathematical proof of the original constants. While the mathematical in terms of appropriated variable definition would suggest that 2 does not equal 1, it fails one small test, it does not prove that 2 does not equal 1 at all. If you evaluate the problem in terms of it's constants, the problem not only violates no mathematical law, it proves what mathematically suggests is invalid, is in fact fallacious. The variable does not predispose the constant, the constant predisposes the variable. It is invalid to discern an answer is incorrect because it's constants do not hold to rules defined in variable terms. It is the burden of the variable to prove it's validity in constants, not the other way around. It is natural to question the previous assertion and suggest that the previous statement is false as a=b does in fact allow for substitution. I agree with that sentiment, but it is not allowable in the respect that it redefines the argument. You can see where this argument breaks logical law when simply reapplying it to the original definition of a=b. If it is valid to substitute a for b and b for a, then (a+b)=(b+b) is a logical argument. Let’s then substitute all terms for their respective values. This results in (1 + 2) = (2 +2) or (3 = 4). This does not disprove the argument of a=b or specifically 2=1, so let’s try something different. Let’s evaluate the same argument of (a+b) = (b+b) but substitute all a’s for b and b’s for a. So (b+a) = (a+a) or (2+1)=(1+1) or (3=2). Obviously that argument didn’t correct why(a DNE b). For the sake of being thorough let’s try subtraction, (a-b)=(b-b). So (1-2)=(2-2) or (-1=0). I promise this answer is more interesting than you might initially suppose. Moving on though, let’s now do as we did before and substitute all b’s for a’s and use subtraction as we did before, (b-a)=(1-1). We get (2-1)=(1-1) or (1=0). The significance of what this tells us, cannot be over emphasized, but the different reasons for why this is important will have to be discussed later. What’s important and relevant now is what this tells us about our problem, specifically about its mathematical evaluation.

We’ve gone over a lot, so let’s do a quick review of what we’ve established. We’ve just looked at the famous proof of 1=2 and considered where it is said to be mathematically incorrect. We looked at that same problem in terms of logic, and applied logical framework to the evaluation of 1=2. We then looked at that logical framework and inferred on the mathematics that were used in disproving that 1 could be equal to 2. From there we went a step deeper and looked at why mathematically the conclusion that 1 = 2 was incorrect logically. By doing so we recognized that the reasoning behind the mathematical rules was logically incorrect, as mathematically a false argument was used to evaluate the problem. Considering this we looked at why this was occurring and recognized that logically, it is incorrect to say that you cannot divide (a-b) because (a-b) would be the equivalent of dividing by zero. We then showed that this was not the case since substituting a for b in respect to their assigned values did nothing more than redefine their values. This highlighted the fallacy in the mathematical reasoning as it showed that unless the values of a and b are defined in constant terms, the argument has no basis upon which to be evaluated. We could see that using the prevailing mathematical logic, the evaluation of (a-b) as zero was no more relevant in the time it was applied then in the beginning of the problem, subsequently by introducing the argument that you may not divide by (a-b) as (a-b) is equal to zero is logically false, as it introduces a false argument, and then evaluates the problem’s reasoning as if the false argument was true. In effect, the mathematical reasoning created a problem and then used the introduction of the false argument to rationalize the rest of the problem.
If at this point you are somewhat lost, that is understandable. What we just covered was very technical. Let’s get back to generally understanding what we’ve been covering. From there if you doubt the reasoning, please go back to the technical argument and show where logical reasoning is broken.

What then are we able to extrapolate from the problem above? From my consideration, I’ve reasoned that one of the most important, if not the most important point of the reasoning above is something that might not initially be apparent, but upon consideration, begins to reveal itself. It is impossible mathematically to disprove that 1 isn’t equal to 2 mathematically, if 1 and 2 are rational constants. That is to say, if a is to be defined as having a constant value of 1 and b is defined to have a constant value of 2, it is impossible to prove 1 isn’t equal to 2. The abstract thought that would surely follow anyone seriously considering this notion might lead to the implications of accepting such an assertion. Can it actually be argued that it cannot be proved that 1 is not equal to 2 logically? I challenge you to consider this, and if you’re so inclined to evaluate in any mathematical and logical terms you see fit. What you will find though is that you cannot conclude that 1 isn’t equal to 2 either mathematically or logically without introducing a false argument.

Why is this though? This rationale seemingly defines reason, although why it does so is not readily apparent. The reason you cannot prove that 1 isn’t equal to 2 without introducing a false argument is actually rather simple, by saying that (a=b) and assigning a value of 1 to a and 2 to b, you’ve done something important. You’ve defined 2 arguments and assigned their values as true statements. The reason it is impossible to prove that 1 ≠ 2 is because in order for this argument to be false, a false argument is needed, and by your very definition of a=b or 1=2, you’ve created the framework of two logical constants. You can’t disprove your argument, because your argument is defined as being true. Does it not make sense then, that in order to prove the argument as false, it is necessary to create a false argument in which you might then disprove? Doesn’t this logic become apparent when you consider the framework we just proposed? Consider this reasoning. It is impossible to prove that 1 = 2 is false, if you’ve defined 1 = 2 to be true.

I’m not naïve to the implications of what I’ve just asserted. Naturally there are many questions that one would ask if accepting what we just went over and what this would mean. Let me establish though that I’m quite sure what argument would be used to discount the implications of what we just covered. A rational person might acknowledge the logic of what we just covered, but would counter with an argument that I believe would sound something like this.

“While the logic is self-evident, the framework of rationale lacks what it means to suppose in that it does not allow for evaluation. It is ineffective to support that an argument cannot be evaluated if the method of doing so does not allow for a reasonable definition of the variable. One cannot support a conclusion by the lack of conclusion, and subsequently framing logic in such a way does not lend itself to the reasonable inference of occurrence.”

Now, it would be rather hypocritical of me to create a straw man argument in which to support what I’m implying. I do though believe the argument above though to be a valid one in that it isn’t incorrect in what it implies. It is fair to suggest that one cannot reasonably accept this framework of logical approach to all problems as this approach doesn’t lend itself to any real understanding, rather the opposite. This reasoning doesn’t lend itself to any greater understanding then what the conclusion it evaluates already supposes. That is to say, what can be learned through the pursuant of logical questioning is what is overlooked in the conclusion of other types of logical reasoning. It would be impossible to prove that 1 + 2 ≠ 4 without the introduction of a false argument, likewise it would be impossible to assume that 1 + 2 = 3, without the allowance of certain assumptions that lend themselves to arguments based in fallacious reasoning. The conclusion of 1 + 2 = 3 is by no means irrational, but its supporting logic may in fact be. Subsequently, the conclusions derived from 1 + 2 = 3 may in fact be correct, but no more so then the original logical validity allow. Further, any following conclusions where 3 is either used to argue or validate conclusions is restricted to these same parameters. Logic resulting in rational thought but failing to account for all influencing variables does not rectify fallacious reasoning when further evaluated, rather it compounds it. One might ask if we are to define any rational argument to be truly fully evaluated, how can we do so in good faith, if we accept that we have incomplete knowledge. I have no answer for this question, which is why I believe it is important to put all schools of thought under the microscope to evaluate where logical missteps may have occurred. This is the point of this pursuit, and I hope you see its value.

If we can accept that it is impossible to logically prove that 1 does not equal 2 without the introduction of a false argument, then what conclusions can we rationally support or accept to be true? If you’re asking this question you are of the right mindset. We’ve shown that by defining any argument to be true as we did in evaluating 1=2, we inhibit our ability to identify where it is false. That is why we will not rely on defining the rational constant. Instead we will evaluate what others believe to be so. The implications of this reasoning as you can probably imagine have a significant impact on scientific reasoning. If one cannot logically show that 1 ≠ 2, how is it possible to make any logical inference as this reasoning supposes what is unarguably a major paradox of thought? It is possible to do so, but in order to do so correctly, we must re-evaluate certain aspects of our rationale. Today’s entry actually highlighted where it is necessary to do so. If you’re wondering where, you can see it in the step following the introduction of the false argument in our example above, the notion that you cannot divide by zero. I challenge you to consider rationally why this rule might exist, and while it is acceptable to define mathematical rules by any means one sees fit, what effect might this rule have on scientific reasoning? We’ll go over this next time, but until then I have a challenge for you:

Define Zero as a rational constant. Is zero nothing, the lack of something or the lack of nothing? Can you logically relate all of zero’s meaning in a way that doesn’t predispose fallacy?



Below you will find the truth operators discussed in today’s post, as well as the truth table upon which deduction is inferred.

Truth Operators

Throughout these notes T indicates "True" and F indicates "False"

Negation: - p ("not")

Conjunction: p•q ("and", "intersection") – also p Ù q (T only when p=T and q=T)

Disjunction: p Ú q ("or", "union") – (F only when p=F and q=F)

Conditional: p Þ q ("if p then q") – Also p É q (antecedent Þ consequent)

Biconditional p Û q ("equivalence", "p if and only if q" ) – Also p º q

You can find the related truth table along with multiple examples of its application here:

http://www.simplyquality.org/Logic.htm

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