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For these and other reasons, strong institutions are established where the individuals involved identify the interests and the aims of the institution as their own. Think, for instance, of a soldier who takes up a rifle in the hope of establishing the independence of his people after a long history of persecution. Such individuals do not need to be coerced to fight, or to be well compensated for their services. The fact that they are fighting for the benefit of their people is enough for them to be willing to throw their lives into the balance for the sake of a collective such as a tribe or a nation, stirring up an ardor in their breasts that moves them to acts of bravery and self-sacrifice that no intimidation or promise of pay could elicit. (p. 63) Human beings constantly desire and actively pursue the health and prosperity of the family, clan, tribe, or nation to which they are tied by bonds of mutual loyalty: We have an intense need to seek the material success of the collective. We work to strengthen its internal integrity by ensuring that its members are loyal to one another in adversity, honor their elders and leaders, and conduct the inevitable competitions among them peaceably. And we toil to hand down the cultural inheritance of the collective, its language and religion, its laws and traditions, its historical perspective, and the unique manner in which it understands the world, to a new generation. (p. 74). His extension of self is one where the collective (the nation) is more important than the individual. It is altruistic in the Randian sense. I agree that it is utopian, but not through and through dangerous and bad; some classical liberal theory is essential to Rand (in particular a lot of concern for individual rights). Let's make it more accurate: collectivism is essential to Hazony's theory, where the stability of the nation requires collectivism.

I'm going to need more popcorn. And I should also add a note of thanks, Jose, for your help demonstrating just how well philosophy works.

I finished reading Knapp’s book, Mathematics is About the World. I rate it 5 stars, but with some room for improvement. Knapp barely mentions arithmetic and counting. More about arithmetic would strengthen his thesis that mathematics is about the world. The positive integers used for counting (and zero) form the foundation for the real numbers. Understanding addition and subtraction of fractions call upon the important concepts of unit and transformation, which he does use extensively for different topics – measuring and vector spaces. As an aside, as I have already indicated, mathematics is also about the way we think about the world. Mathematicians “extrapolate” concepts beyond perceptual reality. Examples are complex numbers and matrices with more than 3 dimensions.

I studied set theory in university. I studied group theory and quantum field theory for masters. I’ve studied chaos theory and fractal dimension in my spare time and I’ve read the Emperor’s New Mind, Metamagical Themas, Godel Escher Bach... why is it I have the deepest conviction that although most of these are interesting and useful they are no where nearly as profound and real an intellectual achievement as grasping Objectivism... many years later? I have great respect for so much of what iI learned in academia and I did quite well but I truly am of the belief, and sometimes it shocks me to think it... after a BSc, and an MSc, (and a professional degree which I will not divulge) ... after all of that... I still did not know how to truly think snd know until Rand and Peikoff. I hate to say it but when I hear of successor functions and when I browse a chapter entitled “Building the real numbers” ... from my old set theory text... I can’t help but think something is wrong... and wonder what mathematics could have become if based on Objective philosophy.

Mathematicians, and different mathematicians, mean different things depending on context. The context is either stated explicitly or reasonably gleaned per a given book or article. So, just to narrow down, let's look at just two of the different contexts. (They are different but they support each other anyway.) To avoid getting too complicated for the purposes of brief posting, I'll give only a sketch, leaving out a lot of details, and not explain every concept (such as 'free variable') and taking some liberties with the notation and concepts, and for ease of reading, I won't always include quote marks to distinguish mention as opposed to use. (So this is not as accurate as a more authoritative treatment). So two contexts: (1) General, informal (or informal mixed with formal) discussion in mathematics about natural numbers. (2) Formal first order Peano arithmetic [I'll just call it 'PA' here]. (1) In general mathematics, we might taken commutativity of addition to be obvious and thus a given. Or one might say: "Okay, I'm going to state some truths about natural numbers from which I can prove a whole bunch of other truths, even though they're obvious anyway. The truths about addition I want to mention are: 0 added to any number is just that number. In symbols: x+0 = 0. The sum of a number and the successor of another (or same) number is just the successor of the sum of the number and the other number. In symbols: x+Sy = S(x+y), or, put another way (where 'S' is defined as '+1'), x+(y+1) = (x+y)+1. The induction rule. Now, with those three truths, one of the many truths I can prove, without assuming anything about natural numbers or what they are, other than those three truths, is the commutativity of addition. In whatever way you conceive the natural numbers, as long that conception includes those three truths I just mentioned, then the commutativity of addition is proven true." Notice that we can't do this with the real numbers, because the induction rule does not work for the real numbers. So, for real numbers, we would take commutativity as an axiom (or in set theory, we would prove commutativity from the properties of the real numbers as they are set theoretically "constructed"). (2) PA, as a system, has a formal first order language, with the primitive logical symbols (including '=' as a logical symbol) and certain primitive non-logical symbols. The logical symbols are: Infinitely many variables: x, y, etc. -> (interpreted as the material conditional) ~ (interpreted as negation) and, from '->' and '~' we can define: & (interpreted as conjunction) v (interpreted as inclusive disjunction) A (so that, where P(x) is any formula with 'x' occurring free, AxP is always interpreted as "for all x, P(x)") and, from 'A' and '~' we can define: E (so that ExP(x) is always interpreted as "there is an x such that P(x)") The non-logical symbols are : 0 S + * We define S(0) =1 S(1) = 2 etc. When the language is interpreted: '0' is assigned to a particular member of the domain of the interpretation; 'S' is assigned to a 1-place function (operation) on the domain, '+' and '*' are each assigned to 2-place functions on the domain. With the "intended" ("standard") interpretation: the domain is the set of natural numbers, '0' is assigned to the number zero, 'S' is assigned to the successor operation, and '+' and '*' are assigned to the addition and multiplication operations respectively. And, since '=' is a logical primitive, we assign it to the identity (equality) relation on the domain. So for any interpretation (such that each variable, in its role as a free variable, is assigned to some member of the domain): x+y is assigned to the value of the '+' operation applied to the ordered pair: <the assigned value of x, the assigned value of y>. And x+y = y+x holds in the interpretation if and only if the value of x+y is identical with (is equal to) the value of y+x. So, to answer your question, in the syntax of the formal system itself, nothing is assumed as to what 'x' and 'y' stand for. But with a formal interpretation of the system, 'x', as a free variable stands for some member of the domain and 'y', as a free variable, stands for some member of the domain. And with the standard interpretation, the domain is the set of natural numbers. However, often we tacitly understand that when formulas such as x+y = y+x are asserted, we take that assertion to be the universal closure: AxAy x+y = y+x (abbreviated Axy x+y = y+x) And so, with the standard interpretation, that asserts that addition is commutative. And we prove it from the PA axioms (we only need the three I mentioned in a previous post, which correspond to the three truths I mentioned in this post).

I would like to see a direct quote of Hilbert on that. Hilbert did discuss that, in one way, formal systems can be viewed separately from content or meaning. But that does not imply that in another way they cannot be viewed with regard to content or meaning. Indeed, Hilbert was very much concerned with the "contentual" aspect of mathematics. Granted, descriptions of Hilbert as viewing mathematics as merely "a pure game of symbols", "without meaning", et. al do occur in literature that simplifies discussion of Hilbert. But for years I have asked people making the claim (here moderated to "reliability") to provide a direct quote from Hilbert. And just looking at Hilbert briefly is enough to see that he was very much concerned with the contentual in mathematics. I'm simplifying somewhat, but Hilbert distinguished between (1) statements that can be checked by finitistic means and (2) statements that cannot be checked by finitistic means. Finitistic means are those that can be reduced to finite counting and combination operations - even reducing to finite manipulations of "tokens" (such as stroke marks on paper if we need to concretize). This is unassailable mathematics, even for finitists and constructivists. If one denies finitistic mathematics, then what other mathematics could one possible accept? On the other hand, mathematics also involves discussion of things such as infinite sequences (try to do even first year calculus without the notion of an infinite sequence). So Hilbert wanted to find a finitistic proof that our axiomatizations of non-finitistic mathematics are consistent. So, there would be unassailable finitistic mathematics (which has clear meaning - that of counting and finite combinatorics) and there would be axiomatized non-finitistic mathematics (of which people may disagree as to whether it has meaning and, if it does have meaning, what that meaning is) that would at least have a finitistic proof of its consistency. So, of course Hilbert regarded finitistic mathematics as having meaning and being completely reliable. And, I'm pretty sure you will find that Hilbert also understood the scientific application of non-finitistic mathematics (such as calcululs). But he understood that it cannot be checked like finitistic mathematics; so what he wanted was a finititistic (thus utterly reliable) proof that non-finitistic mathematics is at least consistent. However, Godel (finitistically) proved that Hilbert's hope for a finitistic consistency proof cannot be realized. Regarding looking at formal systems separately from content: Imagine you have a formal system such as a computer programming language. We usually regard it to have meaning, such as the actual commands it executes on physical computers or whatever. But also, we can view the mere syntax of it separately, without regard to meaning. One could ask, "Is this page of code in proper syntax? I don't need to know at this moment whether it works to do what I want it to do; I just need to know, for this moment, whether it passes the check for syntax." So formal symbol rules can viewed in separation from content, or they can also be viewed with regard to content. Hilbert emphasized, in certain context the separation from content, but in so doing, he did not claim that there is not also a relationship with content.

The system of equations: 2x + 3y = 16 x + 2y = 10 can be placed in matrix form and be pictured with 2-dimensional Cartesian coordinates. I wish I could show the matrix form, but I don't know how to do so here. I omit the picture (graph), too. Similarly, a system of 3 equations and 3 unknowns can be placed in matrix form and be pictured with 3-dimensional Cartesian coordinates. On the other hand, a system of higher order, 4 or more, cannot be pictured with spatial coordinates of any kind. Hence, I for one would not describe such a system as "about the world", but rather "about how we can think about the world." Surely, when we start talking about multiplying matrices, we are not talking "about the world", but rather "about how we can think about the world." Calculus, with its concepts of limits, infinite series, infinitely large and infinitely small, we are not talking "about the world", at least the external world, but rather "about how we can think about the world" and/or methodical thought that takes place in our internal, mental world.