merjet

Hilbert per Stephen: “Thus algebra already goes considerably beyond contentual number theory. Even the formula (1 + a) = (a+ 1), for example, in which a is a genuine number-theoretic variable, in algebra no longer merely imparts information about something contentual but is a certain formal object, a provable formula, which in itself means nothing and whose proof cannot be based on content but requires appeal to the induction axiom." Why not and why? Let a = 4. The formula tells me that I can (1) start with 1 dime and add 4 dimes, or (2) start with 4 dimes and add 1 dime. Either way, the result is 5 dimes. Also, if a equals some other integer > 1, then I can (1) start with 1 dime and add a dimes, or (2) start with a dimes and add 1 dime. Either way, the result is the same count of dimes.

I have no plan to do so. The above also indicates how far this thread has strayed. The title is Math and Reality. Mr. Knapp’s book’s title, sans subtitle, is Mathematics Is About the World. I agree it is very much about the world, but think it’s a little more than that. More concretely, Knapp’s thesis is that arithmetic and geometry, especially analytic geometry, pertain to the world. He defines mathematics as the science of measurement. (Analytic geometry and calculus enable indirect measurement.) I think mathematics is a little broader than that, but measurement is a big part. Functions and vector spaces also pertain to the world. His book is not about symbolic logic, mathematical logic, predicate calculus, or finitary vs. infinitary. His book presents an alternative view of mathematics that is very different from formalism, logicism, Platonism, and others. Regarding the philosophy of mathematics schools of thought surveyed here, his is most similar to Aristotelian realism or empiricism. My view is much like Knapp’s.

Try signing in. I don't see the heart icon when not signed in, but do when I am.

Thank you. The quote is not clear or extensive enough for me to accept as proof ‘Hilbert regarded no part of mathematics to be merely a meaningless symbol game.’ You even quoted Zach: “The infinitary part, on the other hand, is not meaningful from a finitary point of view." How did you get an English translation of Grundlagen der Mathematik? According to this page, some of Volume 1 has been translated to English. Anyway, it’s difficult for me to cite Hilbert himself about meaningful/meaningless without a full English translation available. Returning to the SEP entry Formalism in the Philosophy of Mathematics again: "The Hilbertian position differs because it depends on a distinction within mathematical language between a finitary sector, whose sentences express contentful propositions, and an ideal, or infinitary sector. Where exactly Hilbert drew the distinction, or where it should be drawn, is a matter of debate. Crucially, though, Hilbert adopted an instrumentalistic attitude towards the ideal sector. The formulae of this language are, or are treated as if they are, uninterpreted, having the syntactic form of sentences to which we can apply formal rules of transformation and inference but no semantics." "No semantics" means no meaning.

I see the link I made to Kitcher's Hilbert's Epistemology doesn't work. It was only to an abstract anyway. The full text can be seen on JSTOR.org with a free subscription.

Reply to EC. That’s a colorful interpretation. I don’t feel qualified to judge if it’s accurate or not. I’m too literal. That probably explains my fondness for poetry being pretty low, except for limericks. 🙂

Fair enough. I did write, "David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning" twice. The second time was a direct quote from Kline. The first was copied from something I wrote several years ago, and was likely influenced by Kline. I once also wrote elsewhere following the above quote: "The symbols may represent intuitively meaningful percepts or concepts, but they are not to be so interpreted in pure mathematics." That raises the possibility that the symbols are not always meaningless, but only that they should be so regarded at times. I can't remember ever reading anything by Hilbert himself. There is a risk in that, but relying on secondary sources is very hard to avoid due to limitations of time and interest. Page 48 here is Philip Kitcher regarding Hilbert's formalism. Kitcher regards Hilbert as an apriorist. I think that Hilbert's epistemology has some bearing on Hilbert's view of meaningful/meaningless. Do you agree with that? This referenced by Kitcher might be an interesting read. I didn't find the full article anywhere. Maybe we can more agree on our views of Hilbert's view of meaningful/meaningless if you will cite Hilbert himself or at least a secondary source you judge to be better than, or as good as, Kline and Kitcher. Maybe you have relied on Hilbert himself or a secondary source with the distinction you make between finitary and infinitary mathematics. I won't quibble with what you have said about that.

Of course, Formalism designates a group of people who don't all hold identical views. Hilbert was a major figure of the school. I suggest a more charitable reading of Kline and Knapp. "Formalism -- a major figure being Hilbert -- holds that ....." Note that in both quotes I gave from Knapp's book, Knapp does not even use Hilbert's name.

You might find an article, Imagination and Cognition, that I wrote for Boydstun's journal Objectivity of interest. The topic of memory and its connection to imagination appears several times, including by Aquinas and Hobbes. Oh my, 28 years ago. I won't take all the credit. Stephen was a very helpful editor.

To wit: "Colorless green ideas sleep furiously" - Noam Chomsky (link) "Colorless green" is contradictory. So try another. Short green ideas sleep furiously. 😊

AOC strikes out ProPublica Targets TurboTax Again #3 Spheres of Justice #10

I received Mr. Knapp's book, Mathematics is About the World. It includes 'Hilbert's Game of Symbols' in the subtitle, but doesn't have much more in the body. "At some point during my college freshman year, I realized that neither mathematicians nor philosophers of mathematics shared my perspective, offering only the alternatives of formalism (a game of symbol manipulation), Platonism (a separate world of mathematics), or, as a third, the Fregean view that mathematics is a branch of logic. I could accept none of these choices" (p. 10). Hilbert was a Formalist. "My specific concern will not be with counting objects, but with using numbers to measure magnitudes, such as length, weight, and speed. In this, we should not be surprised to find that our usage of numbers is indeed correct. But we will find that characterizing exactly what we are doing when we apply numbers is not as straightforward as one might have thought. Yet in laying this process bare, one creates the foundation for a similar understanding of mathematical concepts whose relationship to the world we live in may be far from obvious. It is the lack of such understanding that has led to the widespread false alternatives that mathematics is either a formal game played with symbols, a system of deduction from carefully chosen axioms such as the axioms of set theory, or an insight into a Platonic universe of mathematical concepts. On any of these views, the applicability of mathematics to reality must be viewed as a happy accident" (p. 101-2)

This page might help on terminology. My suggesting it doesn't mean I recommend "proposition." The word "declarative" might help. I commend StrictlyLogical for attempting a very challenging task.

Maybe he didn't write it but spoke it. In any case, Kline referring to Hilbert in Mathematics: The Loss of Certainty wrote something quite similar. Maybe Hilbert did not intend it apply to all mathematics, but only part of it. As an aside, there is this.

You might want to modify this after considering fiction.

Great mathematicians are capable of bad philosophy. Two examples are Descartes and David Hilbert. "Mathematics is a game played according to certain simple rules with meaningless marks on paper." - David Hilbert Link. Maybe Hilbert was trying to be funny, but there is plenty of room for doubt. Morris Kline in Mathematics: The Loss of Certainty said about Hilbert's view: "The most reliable way to treat mathematics is to regard it not as factual knowledge, but a purely formal discipline that is abstract, symbolic and without reference to meaning" (page 247). Also, note the subtitle of Knapp's book.

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.

Thanks. I ordered the book from Amazon and expect delivery in 3 days. I will try to put any comments I have upon reading it in the Math and Reality thread.

😊 I must confess that my recollection is not as good as it might seem. My last post was copied from something I wrote a few years ago and then edited a little.

A few years ago I purchased and listened to Pat Corvini's two sets of lectures on number:1. Two, Three, Four and All That; and 2. Two, Three, Four and All That: The SequelThe main topic is her view of numbers. A lesser topic is criticism of Cantor's claims about infinite sets, and his method, with she calls postulational and contructive. Corvini does not say so, but the postulational/contructive philosophical view is epitomized by the famous mathematician David Hilbert's opinion that the most reliable way to treat mathematics is to regard it not as factual knowledge, but as a purely formal discipline that is abstract, symbolic, and without reference to meaning.Her method focuses on "the what" of numbers, whereas Cantor's methods focuses on "the how" of numbers. She sharply distinguishes between counting -- which use only the positive integers -- and measuring -- whose domain is the real numbers (integers + rationals + irrationals). Cantor's method of one-to-one correspondence blurs the distinction.She talks about Cantor in the 1st and 3rd lectures of The Sequel. The last 1/3rd or so of Sequel #2 and the first half or so of Sequel #3 elaborate her view of measurement. Then she returns to Cantor and the postulational/constructive view of the rational and irrrational numbers. In her view there are two sorts of infinities -- counting (conceptualized by the positive integers) and measuring (conceptualized by real numbers and attained by subdividing). The postulational/constructive method blurs the distinction and treats open-ended construction like a concrete.I much agree with what she says, but believe there are even stronger criticisms of Cantor's nonsense. At one point in Sequel #1, Corvini talks in terms of 2-to-1 correspondence, but not any wider range of multiple-to-1 correspondences. Nor does she utilize part-whole logic to criticize Cantor's nonsense.

Can you elaborate? Is there an online source for reading more of what she said?

I made the following comment on the Charles Tew thread: “In another video he talks about mathematics. I don't remember which one. He prefaced his remarks with his not being a mathematician, but what he said about math did not sit well with me. As I recall, he said mathematics is about reality. Yes and no. It is also about our concepts. Show me a matrix, differential equation, integral or complex number in reality that wasn't written by some human being, then I will reconsider.” Link. MisterSwig suggested privately that I say more about this in a separate thread. Before I do, what was it that Charles Tew said that did not sit well with me? The video is ‘Sam Harris Doesn't Understand Math‘ (link). Tew launched a tirade on Sam Harris' saying that the Probability{Jesus will come back in Jackson County, Mo.} < Probability{Jesus will come back somewhere}. Defending that claim when the interviewer challenged him, Harris said it is a mathematically precise statement. In addition to the title he gave the video, Tew made several other comments, including the following: - Harris is disastrously wrong about his Jesus claim. - He said that mathematics is about the world. It applies to reality. - He claimed Harris had a Platonic understanding of math. - Mathematicians don’t know what they are talking about, because they aren’t philosophers. Ditto for physicists. Sam Harris was a little imprecise. He should have said, simplifying, Pr{Jesus to Jackson County} <= Pr{Jesus to somewhere}. Not “less than”, but “less than or equal to.” Even holding that both probabilities are zero like Tew said, the “less than or equal to” formulation is true. Also, more generally, Pr{A} <= Pr{B} if A is a proper subset of B. That is the mathematical principle Harris appealed to, even if he didn’t say it wholly correct. In my opinion, Pew’s asserting that Harris doesn’t understand math is a gross exaggeration. Not even arithmetic and some mathematical probability? That’s all I will say about Harris. I move on to some of Tew’s claims. First, his assertion about mathematicians and physicists is pompous and insulting to many people, quite a few I know personally. Next, is mathematics about the world? Maybe Tew took that claim from the title of the book by Robert E. Knapp. Mathematics is About the World. Nevertheless, Tew's understanding of math seems to me pretty shallow. I have been aware of Knapp's book for a while but haven’t read it. Anyway, in my view mathematics is also about the ideas we use to describe the world quantitatively. Let’s start with arithmetic. Is arithmetic about the world? Mostly yes, but not entirely. Consider 5 – 2 = 3. That’s true for all things countable. But what about 3 – 5 = -2? If I begin with 5 dimes in my hand and remove 2 of them, 3 dimes remain in my hand. However, beginning with 3 dimes in my hand and removing 5 of them is impossible. On the other hand, if the temperature is 3 degrees Fahrenheit or Celsius and then falls 5 degrees, saying it’s then - 2 degrees is valid and about the world. Pure math is an abstract discipline, so mathematicians usually ignore exceptions like not being able to remove 5 dimes from my hand when there are only 3 dimes there. By the way, when negative numbers were first considered, they were regarded as fictitious or false. The algebraic equation x^2 + x – 6 = 0 has two solutions (roots), x = +2 and x = -3. Moving on, a parabola (or circle, or ellipse, or hyperbola) is based on a conic section. The algebraic equation for one can be expressed in Cartesian coordinates – invented by Descartes -- or polar coordinates. Similar for volumes such as that of a cone, cylinder, or sphere. Is such mathematics about the world? It surely is. Cartesian and polar coordinates have a direct correspondence to the real world. The axes in both can correspond to distances in the real, external world. On the other hand, there is another coordinate system which has no such direct correspondence to the real, external world. It is often called the complex plane and is pictured here. The horizontal axis is for real numbers, but the vertical axis is for imaginary (or complex) numbers. We can’t use imaginary numbers to express distances in the real world. On the other hand, imaginary numbers are used to describe the real world in physics, more specifically quantum mechanics (link). The same complex plane coordinate system is shown there again. When imaginary or complex numbers were first systematically explored by Euler, Gauss, and Hamilton more than 150 years ago, their practical use was unknown. So one could say that imaginary or complex numbers were not about the world then. Times have changed. A practical use of them was found many years later in quantum mechanics. So one could say that imaginary or complex numbers are about the world now. By the way, I earlier gave an algebraic equation that had two real solutions (roots). Here is an algebraic equation that has no real number solutions (roots): x^2 + 4x + 5 = 0. The two solutions (roots) are -2 + i and -2 – i, where i is the imaginary (complex) number equal to the square root of -1. To be continued in another post(s). I will say something about matrices, calculus, differential equations, maybe more.

Trump’s Health Insurance Changes #2

I observed a few minutes of a few videos here. https://www.youtube.com/channel/UC8iOCGZj09rvCXhXeya4vkw My impressions were not favorable. In one video he called the session with Jordan Peterson at OCON 2018 a disaster. I was there, and the audience surely didn't judge it a disaster. His Why Socialism Fails based an an analogy with computers was poor. Hayek's explanation was far better. In another video he talks about mathematics. I don't remember which one. He prefaced his remarks with his not being a mathematician, but what he said about math did not sit well with me. As I recall, he said mathematics is about reality. Yes and no. It is also about our concepts. Show me a matrix, differential equation, integral or complex number in reality that wasn't written by some human being, then I will reconsider.