From Britannica:

The Democratic Party has changed significantly during its more than two centuries of existence. During the 19th century the party supported or tolerated slavery, and it opposed civil rights reforms after the American Civil War in order to retain the support of Southern voters. By the mid-20th century it had undergone a dramatic ideological realignment and reinvented itself as a party supporting organized labor, the civil rights of minorities, and progressive reform

In a nutshell.

Do you think you understand Wittgenstein? — Joshs

From ChatGPT:

Overall, Wittgenstein’s profundity lies in his ability to challenge and expand our understanding of how language functions and how it shapes our experience of the world. His insights continue to provoke thought and debate, making his contributions to philosophy both deep and enduring.

and

Wittgenstein's ideas have influenced various contemporary philosophers and mathematicians who are interested in the foundations of mathematics, the nature of mathematical truth, and the philosophy of language. While his impact is more philosophical than technical, it has contributed significantly to ongoing discussions about the nature and practice of mathematics.

I admit, it's been sixty years since I have read anything by the man. At the time I was most interested in his impact on mathematics. However, this was about the time I was taking my one and only course in foundations (naive set theory), and was rapidly losing interest in the subject.

I am not a fan of Wittgenstein's philosophy as it seems to make common sense notions into philosophical "strokes of genius" — schopenhauer1

Ditto. A war hero, yes. Otherwise my eyes glaze over quickly. Early in my mathematical career I tried reading him but found little to interest me.

(2) There are no set theory experts in this thread (or, to my knowledge, posting in this forum). — TonesInDeepFreeze

That is all right. You are enough good to teach me a few things in set theory. — MoK

Yes, and are quite knowledgable in this regard. I never studied anything beyond naive set theory and I have appreciated reading their posts, both instructional (and at times, argumentative). As for reference books, I am compelled to mention one that I have found wonderfully informative and written by a colleague from Colorado College: Introduction to Topology and Modern Analysis (George Simmons)

Continuity is perhaps best approached through elementary topology. Here is what Simmons says:

In the portion of topology which deals with continuous curves and their properties, connectiveness is of great significance, for whatever else a continuous curve may be it is certainly a connected topological space.

Of course, in the case of the reals the previous discussion concerns connectiveness.

Do you think the cannon was a protest or a celebration? Presumably, it didn't have a ball, but was loaded blank? — Ludwig V

There were several Civil War cannons on the lawn of the old armory where the ROTC had made its home since before that conflict. I'm certain one fired a blank at that time. It's funny but I cannot find any reference to it in the media. But I think it would have been a celebration. Most people at the University disapproved of the Klan, and there had been some speculation the KKK might get ugly, but they backed off and were more or less silent.

I had an older cousin who lived in the country and was a member of the Klan. He had worked at some menial job and kept dogs for coon (racoon) hunting. His kitchen sink had a manual pump and chickens ran loose in his dirt yard. When I went out to visit him in 1963 with my fist wife I was astounded in the transformation. He now lived in a nice brick home and when we were met at the door he was neatly dressed and introduced us to one of his best friends, a black man who worked with him at the BFGoodrich plant nearby. The last time I had seen him was in the late 1940s when I was a child. The plant had opened in 1946 and he was still living the rustic life and a member of the Klan then.

Does an equation exemplify symmetry? — ucarr

More or less. Here is where you find reference to the term.

The second law of thermodynamics leads directly to Gódel’s Incompleteness.
Perhaps l should look at dark: matter_energy through this lens — ucarr

And off he goes, where he is led nobody knows. :smile:

Imagine a math space such that : 6+9 =/= 9+6; semi-symmetrical mirroring. — ucarr

A non-commutative operation. Lots of them in math. For example, function composition in general: f(g(z))=/=g(f(z)). Also called non-abelian.

I can't really talk about the Dems, but I have the impression that the Dems, back in the day, were an alliance of (mainly social) liberals and political left wingers — Ludwig V

Far from it. I grew up in a segregated South and the Democratic party supported that. Political winds finally shifted during the 1960s.

From Wiki:

During this period, the white-dominated Democratic Party maintained political control of the South. With whites controlling all the seats representing the total population of the South, they had a powerful voting bloc in Congress. The Republican Party—the "party of Lincoln" and the party to which most blacks had belonged—shrank to insignificance except in remote Unionist areas of Appalachia and the Ozarks as black voter registration was suppressed. The Republican lily-white movement also gained strength by excluding blacks. Until 1965, the "Solid South" was a one-party system under the white Democrats.

I was in a math class at the University of Alabama in 1963 when Governor Wallace was asked to step aside and allow two Afro-American students to enroll. He complied and those of us on the sidelines cheered. An old Confederate cannon went off at the time, but I can find no reference to that.

My first vote for President was the 1960 election, and I caste my ballot for JFK. He had been a genuine war hero, and when he extended my tour in the USAF for a year because the Berlin Wall was going up I forgave him. Turned out it worked out well for me.

There doesn't seem to be anything about the races for the Senate and the House. But isn't it just as important as the Presidency? — Ludwig V

I'll intrude if you don't mind. The answer is yes, indeed. Here in southern Colorado our representative to the House, Two Gun Lauren Boebert, has moved north into another district. We do have a decent Republican candidate, but I have seen and heard nothing about him. The Democrat running is more in the old fashioned mold and will receive a lot of conservative votes I suspect. Nevertheless it will be one less Repub in the House - and the ratio is tight.

Issues involving new allocations of money begin in the House, so control there is critical.

and I, apart from a mathematical background, align on political matters it seems to me. I am still a registered Democrat, but it has been awhile since I have thought of myself as one.

Do you really not understand what a domain of a function is? Or are you trolling me? — TonesInDeepFreeze

Does make one wonder, doesn't it? :roll:

I am a retired physicist and my knowledge of mathematics is very rusty due to my age — MoK

Don't feel bad. I"m a very old retired mathematician and have had to look up filter trying to understand @sime 's comments.

E.g, one can simply define a "line" as referring to a filter, — sime

Hmmm. Care to explain? (I recall having difficulty with filters, ultra filters, etc. in grad school a half century ago. I only encountered them in passing - not in my specialty area)

The really basic question is why there is no decent candidate on either side — Ludwig V

Driving the other day a car revs up behind me while I am going a little over the speed limit, then barrels around me into the oncoming lane on a curve. I wonder if the answer to your question has something to do with a general lack of patience. Everything has to be done as quickly as possible it seems. Patience is no longer a virtue. Just a thought.

What do you think of the connection that Schlesinger makes between entropy and Godelian incompleteness in "Entropy, heat, and Gödel incompleteness" (2014)? — Tarskian

If the dynamics of a system becomes so complex that G¨odel incompleteness
prohibits a complete description of its dynamics, the necessary information
– to determine the dynamics – is fundamentally lost on a universal Turing
machine.

Interesting observation. I'm not sure it takes G-incompleteness to reach this point. Dynamical system structures may be just too complex to handle without going into the realm of uncountable math garbage.

Are there tricks or strategies that the ordinary person can do to improve mathematical ability?

Devlin knows a lot more about this than me, but In all my years I haven't witnessed any kind of improvement that hasn't come from simply picking up an elementary math text and making an attempt to understand it. Or taking an elementary class. With Wikipedia as a sort of backdrop it's easier to do this these days.

Sometimes people convince themselves they have little to no math ability. Then it's really hard to make progress.

The same question arises with critical thinking. I am discouraging there also. But I would love to be shown wrong. ChatGPT disagrees with me, but its suggestions assume someone who has certain personality qualities. Can these be cultivated?

is there a logically sound argument claiming there is a causal relationship between entropy and incompleteness? — ucarr

Possibly between entropy and incompleteness in its more traditional meaning. Not with the math variety.

Too much of a stretch for my declining imagination.

The weird thing is I am fascinated by math. I have books and DVD's about math. I want to learn the language of math and I understand learning a language is one way to keep our mental powers as we age — Athena

You might not be aware that the infamous western gunman, John Wesley Hardin, when in prison worked his way through an algebra textbook. He also became a lawyer.

In one of my sets of college lectures, the professor can talk about knots for at least an hour. — Athena

That would have put me to sleep. :cool:

It appears all you have shown is the distance between consecutive means tends to zero. The last sentence is a little weird. The previous sentence says it all if one takes a limit.

I haven't kept up with this nonsense. Kudos to those knowledgeable who have.

I gather that the numbers were down and have gone up since. I don't know why — Ludwig V

Surge the border

This might be a clue. I am enjoying the discussion by the two of you. Better by far than what is found on the visible pages.

Sorry to hear of the COVID. Get well. :smile:

To be Christian, you need to believe that Jesus Christ is divine and died for us — Lionino

Unitarian Christians?

the set of all quantities is countable, as is the set of points on a number line. — hypericin

Not so, my friend, if we speak of the real number line. This has been chewed on on this forum until there is little left to be said.

Paywall. I'm surprised people still go to movie houses to see a film. If one categorizes and focuses on westerns, for example, I suspect the Clint Eastwood spaghetti series might qualify. Just me. And not Hollywood.

Here is an excerpt from a Max Planck Institute article :

For example, the historian Ingeborg van Vugt has used this multi-layered approach to explore the different ways in which information circulated in the Republic of Letters, the long-distance intellectual community of the late 17th and 18th centuries in Europe and America. Such research allows us to better visualize how the Age of Enlightenment, driven forward by these intellectuals, developed. The next step could be to statistically model this network, and so be able to pursue her research question by integrating an even broader wealth of data.

A network model for studies in history of knowledge has to consider an unusually varied set of data. There is the data of a social nature concerned with people and organizations; that concerned with material aspects of history, such as the conservation life of a book; and the data that represent the actual knowledge, the content of the sources. These are three different levels of one and the same evolving network for which explanatory mathematical models have been rarely conceived and even less realized. From this perspective, history writing is even about to challenge applied statistics.

A dynamical systems approach is an enormous elaboration of the simple notion of composition of functions in which there is a mechanism of feedback that can help direct the process level to level. But even this germ of a system can be complicated. Add a huge reservoir of data and progress depends upon interpreting what it all means.

I suppose I am still a Christian, though I have not been to church for a long, long time. Thinking of Jesus as a man makes no difference. His teachings are important. So, number 4.

What if we did not use words, but communicated with math? I know mathematicians can do that, but what if from the beginning we all did? I am sure my IQ would be much higher if I could do that. And I wonderful how thinking in mathematical terms might change our emotional experience of life. — Athena

Interesting idea. Logicians might be able to do this, but math people use words and symbols. I have never heard of a math research paper written in math symbols only. Thinking in mathematical terms is common amongst my colleagues, but even there one talks to oneself with words.

I think that the vast majority of academic papers are considered to be irrelevant. In that sense, it does not matter if the justification supplied is solid or not. Nobody cares anyway — Tarskian

Pretty much the case in mathematics. One result is that even competent referees skim over details too often, especially if the author is a respected academic. Lots of mistakes are published, mostly non critical.

The mathematics of this is precise. A fractal distribution system has a log/log or powerlaw scale of size. That is how a geography can be efficiently covered so every drop of water or wannabe flyer gets an equal chance of participating in a well-organised network of flow — apokrisis

Interesting reference. A parabolic fractional distribution is a little obscure with the Wiki page getting only 7 views per day.

Thanks for the info. :cool:

The nearby city of Pueblo (111,000) was given a government grant to re designate several two way streets into one way with a bike lane. For the past six weeks I drove these streets to medical appointments each weekday, and saw a total of three bike riders.

It is where pure mathematics tries to establish a foundation of knowledge that I am disgruntled. The effort is laudable - but mathematicians have gotten themselves stuck in a dead end and appear unwilling to extricate themselves. — Treatid

It is not a popular function of "pure" mathematics to delve into these issues. For example, arXiv.org lists the number of math papers submitted in this last week: Total-783, Logic-5, History&Overview-1. To compare: Category Theory-18, Complex variables-18. Others-751 (29 additional categories).

Interesting comments. For me, much stems from density of life. My daughter lives in Brooklyn in an area having roughly 36,000 inhabitants per square mile, while I live on the high prairie of southern Colorado in a ruburban environment of about 400 per square mile. The necessity of private transportation to some extent is a function of this statistic.

Years ago I was on a train going from London to Wales, passing through lovely meadow lands, when I saw a large vertical building rising abruptly in the distance, surrounded by a high wall. An apartment building made to cram many into a relatively small space. All around were fields empty with the exception of a few cows and trees.

High density or low density. A largely political argument. I saw a cartoon recently that showed differences between Democrats and Republicans, and for housing the Dems favored the high rise building, cramming as many as possible together regardless of ethnicity or religion or political persuasion. The Republican model was a cottage in a yard filled with grass and flowers with a cute white fence surrounding it.

That is why I have personally never treated and will never treat philosophy or mathematics as more than just hobbies — Tarskian

Understandable. Many others likewise. For the last 24 years math explorations have been a hobby for me as well.

Mathematics does not have direct practical applications, mostly by design so. That is often a good thing, but it also means that the academic consensus has much more weight than it would have, if there were practical applications — Tarskian

There's just the mutually back-patting consensus, or else meaningless grades on a collection of otherwise irrelevant tests and exams, or even the eternally back-patting citation carousel. That is why I have personally never treated and will never treat philosophy or mathematics as more than just hobbies — Tarskian

Your opinion has been noted. Actually, I agree with the first statement above. The second sounds a little bitter.

, Thanks. But way, way outside the scope of vector spaces I ever encountered, like spaces of contours in the complex plane (where I still dwell).

Not sure if mathematical logic is just a curiosity — Tarskian

To make your point would require some sort of poll of mathematicians asking "Are the Foundations of Mathematics important to you as you pursue your explorations into your specialties?"

I'm betting most of my colleagues would say no. I think you are possibly unaware of the enormous scope of mathematical inquiries these days. Look up college catalogues of math courses and find how many have set theory prominently displayed in more than an introductory course. Here are Harvard's Offerings. M145a and M145b and M385 out of how many courses? Plus, of course some overlaps.

When I was somewhat active over 25 years ago Foundations never came up at the conferences I participated in - international groups. Except a joke or two about the continuum hypothesis.

I had not heard of fusion category, but found it has 7 views per day on Wikipedia. And Categories has a little over two papers per day on arXiv.org . These numbers give a very crude estimate of a subject's popularity. If I were a lot younger and healthier I might try to learn something of this topic. Never thought of vector spaces with irrational dimensions either.

One could say the crisis is still going on, as we don't know whether ZFC is free of contradictions (and perhaps never will). — Lionino

I suppose it is, especially among foundations mathematicians. But I would not say it remains a crisis within the broader scope of the profession. Mostly a curiosity.

You have your basic facts all wrong. — fishfry

:clap: :cool: