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:

We could spend decades arguing back and forth over whether mathematicians are applying rules consistently to the staircase paradox — Treatid

I don't think you would find a mathematician who would spend more than hour on it.

To me, "shake on it" signifies agreeing to a course of action rather than agreeing on a belief. If you meet someone and ask if they too believe in Kamala Harris and they say yes, do you then shake on it? I would not.

(1eg) If a theory explains an observation, then the theory is evidenced. — Hallucinogen

What is that flying across the sky leaving behind it a trail? It must be Icarus on his way. Yes, my theory is evidenced! :roll:

I actually did not invent the term "foundational crisis of mathematics" by myself — Tarskian

OK. Thanks for the links. It should be emphasized that the crises is in the philosophy of mathematics. Mathematicians by and large ignore the crises (unless they are into fundamentals). :cool:

Here we go again, assuming a stroll along an uneven path is the same as wandering through a minefield. — jgill


So, the idea is that the use of Godel numbering in a logic expression points to making use of the philosophical capability of the language and therefore turns the expression into a philosophical one. There may be exceptions, though. — Tarskian

I was just commenting on your referring to "a foundational crises in mathematics". I doubt many mathematicians would agree there is a "crises". Concerns perhaps.

The origin for what I write, is of course, the foundational crisis in mathematics — Tarskian

Here we go again, assuming a stroll along an uneven path is the same as wandering through a minefield.

How is all this relevant for defining philosophy? How is this the relevant to philosophy in any way? — Ludwig V

:up:

Mathematicians have a long career of coming across inconsistencies and hurriedly changing the rules so that this particular inconsistency no longer counts. — Treatid

In the case of the staircase paradox mathematicians simply accept the fact that the sequence of arc lengths does not converge to the length of the arc that the sequence converges to under the supremum metric on a space of contours. No changing of the rules.

Mathematics has a massive foundational crisis with insurmountable issues. — Tarskian

That is why I find the foundational crisis in mathematics an exhilarating subject — Tarskian

Well, it's good someone is interested. :roll:

Nothingness was abhorrent ? Geometry? Don't know and don't care. :cool:

Math originally came from accounting, believe it or not — frank

Well, here is what ChatGPT has to say:

Mathematics and accounting are deeply intertwined, but mathematics did not originally come from accounting. Instead, mathematics has a much broader and older origin that spans various domains.

Here’s a brief overview of how these fields are related:

Early Mathematics: The origins of mathematics date back to ancient civilizations such as the Babylonians, Egyptians, and Greeks. Early mathematics involved basic counting, measurements, and arithmetic. These practices were crucial for various practical activities like agriculture, trade, and construction.

Accounting Origins: The practice of accounting, especially systematic bookkeeping, has roots in ancient civilizations as well. For instance, the Sumerians developed one of the earliest known accounting systems around 3000 BCE, which involved recording transactions on clay tablets. Accounting was essential for managing resources, trade, and taxation.

Development of Mathematics: Mathematics evolved from these practical needs into a more abstract and systematic study. Ancient Greeks, such as Pythagoras, Euclid, and Archimedes, made significant contributions to mathematics that went beyond mere accounting and measurements, exploring geometry, number theory, and more.

Interconnection: As mathematics developed, it increasingly influenced and was influenced by accounting practices. For example, the development of algebra and calculus provided tools for more sophisticated financial analysis and modeling.

In summary, while accounting and mathematics are closely related and have influenced each other, mathematics as a discipline predates accounting and encompasses a much broader range of study than accounting alone.

In mathematics - a paradox (inconsistency) demonstrates a faulty set of axioms — Treatid

Not necessarily. The Diagonal paradox can be extended to a sequence of smooth curves that converges to a limit curve in the complex plane in which the disparity of lengths is infinite. There is no argument I have heard of that implies fundamental axioms of the real (and complex) numbers is at fault. I seem to recall Aristotle was aware of this discrepancy of lengths.

Is there a question here? There is a lot more to science than honest bookkeeping.

edit: didn't see the previous replies.

When I donate to Wikipedia, in a sense I receive a benefit indirectly and help provide benefits to others. In a way altruistic philosophy I suppose. When I transfer money from one savings account to another at the same interest rate its as though nothing has happened Unless the second account has an additional name on it.

Describe why arbitrary transfers are philosophically significant — Mark Nyquist

Donations can shape society. Simply moving money around usually does not.

This thread is a stretch. :roll:

Abraham Robinson's definition of h revolutionised mathematics in the 1960's. — alan1000

This is an exaggeration. There are probably universities around where this is taught regularly, but it has not caught on to any significant degree in general. A colleague of mine who taught at the U of Colorado told me they made an attempt to start a course in the subject, but it flopped. I don't see any course in their curriculum now that focuses on non-standard analysis. But there are courses in foundations where it may crop up.

So, rather than drift off into systems that depart from the standard material on the real numbers, its best to stick with the widely accepted ideas. Just my opinion.