The Logic of Mathematics

Mathematics, has never seemed too logical to me and this is due to (1) pedagogy, and (2) the use of the "equals" sign. 

   Now for pedagogy, most math teachers or at least their students, seem to conceive that math is about real objects -marbles, pyramids, train tracks, etc. and that math is about countable things. However most mathematicians and aristotle would state that math is just symbols made up by the mind and arranged logically with only an incidental application to reality. This response obviates some of the obstacles to learning math that the "realist" view builds up; for instance, that negative or imaginary numbers don't exist and yet math deals with them. Additionally it also makes the theory of equations much more sensible since equations are not like see-saws where, by the operation of weight, each side of the saw tends to balance w/the other side but are rather logical relations of equality such that if one side changes, then the other side must, logically, change. A non-realist view also has the advantage of circumventing the biggest problem in realist math: simultaneous determination; for in mathematics the parts of an equation do not cause the other parts of the equation or expression. 1+1 does not cause 2 in the sense that a man throwing a ball causes it to fly, for math is not physics. So mathematical objects have at best, only a mental causal relationship, and the inputs and outputs of expressions are always stationary since every way one manipulates an expression makes it either equal to itself or it completely changes the definition of the expression so that no matter the expression or the changes done to it, that expression is always itself. This law of identity is always and everywhere the same as the law of simultaneous determination.

   The second reason is that when one solves an equation one uses the fact that it is equal to something in order to prove that it is equal to some other thing. However, because of the use of the equal sign, everything is explicitly stated to be equal prior to the proof that everything is. And so, to solve an equation is basically the same thing as to reason circularly. But circular reasoning is illogical so that raises a question, "is it possible to base mathematical proof (equation solving) on a logical foundation?" The answer to this is yes for several reasons but let me just sketch out my preliminary conclusions first.

1,2,3, etc. are defined like this: 1=--1, 2=--2,3=--3 or each number is the complement of its complement.

3-1 is just the difference b/t the complement of 3’s complement and the complement of 1’s complement. And this is just the complement of the set “3&1” in the set of 3 things. And this is 2.

Clearly I intend to use set algebra to deduce normal algebra. But set algebra is basically logic, so normal algebra will be established logically. 

A quick post on my feelings RE: Austrian Economics

Intro: though some said that Austrian Economics is a defense of economic liberalism -that is,a defense of immoral economics -I say contrarily.The below is my reply in the debate:


"AE (Austrian Economics) is just a science of action qua action. My remarks will have to be short and my tardy appearance in this debate will make them shorter still, so forgive me for the cryptic-ness.
   Essentially, AE uses ethical-sounding terms, just as the chemist does. For the latter, “ideal gas” doesn’t mean “a supreme standard of human life that is gaseous in form” but it means “a powerful gas qua gas”. So “fair price” in AE means “a stable price qua price”. AE also defends liberalism but this is more historical accident than anything else. If an austrian economist ever said that free-trade was good, he either meant that free-trade was a stable condition qua trade, or he meant that free-trade was the best way to get wealth (all things equal), assuming people want wealth (even though it may conflict with justice). In no way does AE preach capitalism as a matter of necessity -as a matter of necessity it preaches nothing but analyzes everything.
  So I think there are definitely some things to be learned from Mises/Rothbard -just as there were important things to be learned from Aristotle even though Plato was long-preferred to the former."

Philosophical Bloviations: Of the Relation Between Faith and Reason Pt. 1

  Yes, it has been a while since I posted on this blog but there has not been a like absence of philosophizing on my part. Today I want to write about the relation between faith (supernatural theology) and reason (natural theology) from the point of view of St. Thomas Aquinas's Summa Theologica.
   Now let me establish some necessary premises about Philosophy (science) and its relations with reason and about Supernatural Theology, before I analyze them later. First, all men shouldn't know what is above the knowable ("what is above reason"). To be unknowable is to be irrational -not approachable by ratiocination. Now nothing irrational is in philosophy. Further, nothing is known but what is true, that is, if something is known then it is true for other wise it would be possible for some known thing to be outside the set of true things, which contradicts the idea that nothing is known accept what is true. And again, knowledge is only concerned with being so that if it is knowledge then it is about being (and being=truth). That being equals truth may be seen later when Aquinas writes that all that is, is true. However all that is, is treated of in philosophy. Now because all knowledge is justified, and all justification is ratiocination, then clearly all knowledge is rational. Here is where the difficulty ensues, for if all knowledge is contained in philosophy (and by modus ponens, all philosophy is rational), then it looks as if philosophy will suffice for, and supernatural theology is useless for, all human needs.  And yet, any Catholic believes that supernatural theology is of utmost importance. How does one solve this problem?
   Sacred Science (or S.S. or supernatural theology) is the study of things above reason and accepted on faith. Sacred Science is science because sacred science proceeds by means of faith, and all sciences are differentiated according to the means by which the science's conclusions are made.
   So S.S. is not rational and is based on faith. Now faith= -(ratiocination) and yet S.S. is also true. So clearly, by a fourth figure syllogism, some irrational things are true. Problem number two now ensues:  If philosophy contains nothing irrational, then it must contain something rational by disjunct subtraction. So it is possible at least, that philosophy contains all knowables. But as plato observed, knowledge is true, so nothing known is false. So if philosophy has all knowledge and if all knowledge is true, then philosophy would contain all truth. But again, some truth is irrational, so some philosophy would be irrational which contradicts the fact that philosophy is only rational. So clearly philosophy doesn't contain all knowledge but if that is true, then that would contradict the idea that everything that is, is treated of in philosophy. Deep are the depths that one must plumb, to resolve a weak mind's contradictions. But I realize that this post is lengthy so I'm going to put the left-overs of tonight's mental meal in the fridge and return to the subject tomorrow.

Towards A Deductive Approach to History

   History presents many problems to the historian. Firstly, how does the historian know, that what he knows is true? Second, how can the historian certainly discover past events? And thirdly, how is the historian supposed to do research?  All these questions can be answered only by a recourse to deduction.
   First, what is deduction? Deduction is the method of reasoning opposed to the current mode of reasoning in history. More precisely, deduction is the type of reasoning wherein the truth values of the premises necessarily imply the truth value of the conclusion.  This is opposed to the current inductive method of historiography. Now, the inductive method, is a type of reasoning where the truth value of the premises is related probabilistically to the conclusion. In the first mode, the evidential relations are ones of certainty, and in the second (inductive) mode, the evidential relations are ones of probability. So if your premises are true in a deductive argument, then the conclusion is certainly true. But if your premises are just probably true, then they are inductive, and so the conclusion is only likely.
   It is evident that if history is to be probabilistic, then the historian is faced with the problem of how he certainly knows what he does. The inductive argument, due to its probable nature, is unable to encompass every instance of a class of things. If I say that all doves are white, then this statement could not be an probabilistic one, since the statement is about the certain totality of doves and to know all doves are x, is to have a truth probability of 100%. But to have 100% certainty is against the nature of probable conclusions which can only very from .0001-.9999 integrals of certitude. The contrary is true of the deductive argument. But the whole reason for history is to know the totality of all particular events inasmuch as these occurred. So the inductive method is not a useful historical method, since if a method can fulfill the purpose of a science, then it is useful, but induction doesn't fulfill the reason for history, so it isn't useful by simple modus tollens. But again, if something impedes you in the attainment of a goal, then it is a problem. But induction impedes the goal of finding every past event, so finally induction is a problem for the historian.  But if induction is ruled out, what other choice do we have? In fact, we have only one other choice and this is the method of deduction. Deduction, as was explained, takes certain premises and argues toward certain conclusions. As such, it is possible to deduce statements that apply to all members of a class of things. These statements are called "universal". And since deduction can do this, and since history consists in the aim for the universal, then deduction is the best method for doing history. And for much the same reasons, deduction is also the most useful historical method. Having established this, we have also found the answer to the first problem of the historian, "how do I know what I know is true?" And here we answer that since deduction is a way of knowing, and since it establishes certain conclusions, we can conclude that it establishes knowledge and certain conclusions. But all conclusions are a type of knowledge -since conclusions are known and everything known is knowledge and therefore deduced conclusions are knowledge and further are the best kind of knowledge; they are certainly true statements.
   What I would now like to share with you are some of the fruits of deductive history. Now in teaching and learning, it is meet to proceed from what you know more to what you know less. I personally, know more about American history than anything else, so I begin my analysis from that subject. The analysis deduces several new conclusions from several old facts. The first fact is that all english national projects were executed through the agency of Elizabeth I, the second is that all English merchants were the proximate executors of these projects (Sir Francis Drake for instance), and the last fact is that, some puritans were English merchants. It follows then, that some puritans executed Elizabeth's will.  More controversial theses can be proved presently.The first premise is that all white people didn't like slaves, the second is that some white people were slaves, and so the conclusion is that some slaves didn't like slaves. This is highly surprising because most people tend to see the slave class as homogeneous in both its attitudes as well as its racial makeup. But even now, in high school texts, this idea has been effectively combated for instance, slaves coming from one African kingdom frequently disliked slaves from another and slave-on-slave hatred is proved again from a different premise (if you think that only blacks were slaves). Yet another thesis can be derived about the class-conflicts of American history. For instance, All high officeholders were rich men, all rich men engrossed land, so all high elected officials engrossed land -something that people generally don't see in the character of democratic government, yet such were the politician's actions. For a final controversial thesis, I would argue that black men had important social functions in colonial new England. According to some texts, all new england men were powerful patriarchs who transacted the important work of the society while their wives transacted the domestic work. But some males in new england were black (since there were some black slaves -mainly butlers -who resided in new england), so some blacks transacted socially important work. This and all the other conclusions which have been derived, are absolutely demonstrated and may be called apodicitic knowledge.
   What then, is to guide the historian's research -our last query.  Now no historian can escape having a point a view because if he tried to argue against having a point of view, he would be taking a point of view. But a point of view, taken as a preconceived guide, is essentially a research paradigm. So no historian can escape having a research paradigm. Further every research paradigm should be useful and as we saw earlier, the most useful thing is that which does not impede the approach towards the goal. But as we saw above, deduction is the most useful method. So we know that deduction is useful, however we cannot be sure that it is a research paradigm, even though all research paradigms are useful. So then how do we solve this logical problem? The solution is that since there are only two choices between useful methods, induction and deduction, and since induction is not useful, clearly only deduction exists in the category of "useful methods", at least for history. Hence all research paradigms in history must be deductive in character.

Live Blogging: A Modern Manual of Scholastic Philosophy (chapter 1, part 3)

   Once again I return to the work and pleasure of studying philosophy. In this part, our author indicates that the process of abstraction (of breaking up ideas into its most universal and simplest parts) cannot go on infinitely. At some point an idea is no longer explained by anything else but rather explains all things. When an idea reaches this level it is called a principle. A principle is also called a reason.  One must withdraw in awestruck realization at the method of this treatise since it is, practicing what it teaches while it teaches it -namely abstraction and synthesis. And since anything that repeats is called consistent one must admire the author's consistency. But I digress, continuing on, philosophy is the use of simple reasons to explain everything. The first knowledge that is attained by a human being (that is in his childhood) is spontaneous, that is,  the sense organs are stimulated by natural things and this begets knowledge. When the will controls the other faculties of the human, so as to focus its power on abstraction, and the mind abstracts and then unites the abstracted ideas, then we have formed a particular science. But the particular sciences are also analysis of objects under a special form. But how can something be both "analysis" and "synthesis"? Perhaps a sum of ideas is being subtracted from some larger whole. Also, since this process of analysis and synthesis is frequent, it happens that there are frequently many sciences being created.  The mind however, wishes to unify the results of the particular sciences and to explain them by the simplest principles. Wherefore arises the use of philosophy. But again how can philosophy -synthesis -be an explanan of the simplest principles -analysis?
  All things have three qualities in common, quantity, movement, and substance. These triple objects form the basis for what is called the most general philosophy. Philosophy may be defined as the science of things through its simplest and most general causes.  Or what is the same thing, philosophy is the science of all through the simplest reasons. Philosophy as science is opposed to spontaneous knowledge -and what comes to the same, is opposed to the knowledge of the man of the street. But if that is true, and the very beginnings of all science lay in spontaneous knowledge, then philosophy is opposed to itself which is contradictory. It is also against belief as well as uncertainty. Indeed science implies certainty. St. Thomas Aquinas says that if we have a reason why and how something is, then we have certain knowledge of the thing. Every science gives all the reasons for an object considered from a certain P.O.V. So all science is synthetic. Philosophy regards again, the sum total of objects. The formal object of philosophy is simple while the material object of philosophy is indeed all objects. Philosophy is truly science in the highest degree -science that penetrates all the way to the bottom.
   Several things appears evident from what has been written. First, it seems that if  all thoughts are subject to uncertainty, and all analytical or synthetic ideas are thoughts, then all philosophy (which as we saw is either synthetic or analytic) is open to uncertainty.  Second, if all thoughts are gained through sensual experience of the natural world, and since no two person's experiences are alike, then there must be a gargantuan host of philosophies -at least as many as there are people.

Live Blogging: A Modern Manual of Scholastic Philosophy (chapter 1, part deux)

  I back again, on the same chapter from yesterday, so to immediately move forward, the thing which distinguishes thought from anything else is the simplicity and universality of its ideas. That is, the abstractness, and extension of ideas are necessary conditions of thought and probably are sufficient conditions too. All men have this ability, to consider a thing in separate notions and this ability distinguishes men from animals. Further, this ability is wholly wanting to animals, all intellectual acts are accompanied with it, and it is THE distinctive feature of intelligence. From the fact that all men can abstract, and from the fact that this ability is wholly wanting to animals, I dare say that we have a biconditional statement here implying that man and only man has the ability to abstract. And continuing, since all intellectual acts are apparently abstract acts, and all abstract acts are human, then all intellectual acts are human (but how then are we to explain the intellectual acts of angels? -this we shall endeavor to uncover later). Presently our interests lay elsewhere. To treat further of the idea, the abstract idea is the simple idea and vice versa, and all extended ideas are comprehended if and only if, comprehended ideas are extended ones (that is, comprehension=extension in meaning). There is an inverse proportion between the comprehension of an idea and the simplicity of an idea. For instance, an idea about red tomatoes is applicable to any red tomato, whether it be bruised, soft or hard, etc. But the less simple idea about red small tomatoes is much more specific -hence the more ideas make up a concept (the greater the comprehension) the less the extension (or application/universality). From this I will tentatively state that if an idea is of high extension, then it is of low comprehension, and that if an idea is of low extension, then it is of high comprehension. But is this not contradictory: (-e-->c), & (c-->-e)? How do we resolve this problem if it is a problem? Finally, it is obvious how we must abstract/analyze things into simple ideas before comprehending them into complex ideas in order to understand anyone thing.

Live Blogging: A Modern Manual of Scholastic Philosophy

  As great an admirer of the Scholastics as I would have no choice but to live blog the teachings of this wonderful book by Cardinal Mercier. Let us dispense with pleasantries however, and move directly to the meat of the work.
  The first chapter of the book is entitled, "The Introduction to Philosophy" (also note the work's wide address -although called a "Manuel of Scholastic Philosophy" it addresses itself to both the seminary student and the wider public). So lets begin -the book starts by saying that some argue that the whole field of certain and verifiable knowledge is rightfully monopolized by the special sciences (physics, math, chemistry, history, etc.). Indeed, since advanced instruments have sharpened our perceptions, we have been able to multiply the number of sciences so as to take up the whole of what can be studied. At best what remains of philosophy can only be shadowy or unverifiable fancies. Here though, the author makes an interesting word choice -that is he uses the word analysis. And later we find that analysis means "to break down ideas". So if this is true, the author seems to characterize modern positive science as sciences which primarily break down "synthetic" ideas.  Continuing on, philosophy however does not want to be another science besides the other sciences but it aspires to a place above and after the aforementioned studies. Fascinatingly, the author further describes philosophy as a science which seeks to understand the objects of the other sciences in an ultimate way, inquiring into their relations and connections, and proceeding from thence, to conclusions of universal applicability and to inscrutable ideas (note that even philosophy, is supposed to be analytic, for whatever is inscrutable cannot be further analyzed).  Philosophy is the search for the highest causes of things. More will be said of this chapter later.