Saharon Shelah and number 4, a thread.

Shelah is an incredibly prolific mathematician working mostly in mathematical logic who has (co-)authored around 1500 papers so far. His other superpower is the ability to discover number 4 where it has absolutely no reason to be. 1/32

Example 1. "There are just 4 second order quantifiers".
In first-order logic, one is allowed to form statements about structures, such as graphs, groups, fields,
etc., with only quantification over elements allowed. 2/32

So you can say "exists x, ..." or "for all x, ..." with x ranging over the elements of the structure M, but you can't say "for all subsets of M, ..." or "for all binary relations on M, ...", etc. Allowing such
quantifiers puts us in the context of _second order logic_. 3/32

It was well-known in logic and computer science that one can express more complicated properties by quantifying over binary relations on M rather than just over the unary ones. But can we say even more quantifying over ternary relations? 4/32

And how does this compare to quantifying over functions, bijections, or any other infinitely many possible types of relations on M one can think of? In his 1973 paper Shelah proved that there are exactly 4 (four) possible types of second order quantifiers. 5/32

⬇️⬇️⬇️ The "Math" of Fear

As it is Thursday, I have a post for you today in anticipation of Gov. DeWine's new Maps of Fear which he has tried to phase in. I have previously discussed both the new red and blue maps and their issues -

- but today I hope to illustrate even more clearly what level of manipulation is going on, particularly with the new blue 'ICU Utilization' map.

Attached I have two images.

The first is a hyper-simplistic (and unrealistic) but illustrative example of the calculations that the new blue map goes through to arrive at its numbers.

Instead of telling us the percent of beds occupied by COVID+ (not necessarily ill *with* COVID, but testing positive) - the governor and ODH only calculates the percentage of COVID+ out of just the *used* beds.

In the first image, I have an example of a 100 bed ICU with just two patients - one COVID positive (red) and one COVID negative (green). The other 98 beds are empty and able to accept new patients.

When I was starting out as a grad student, I remember being frustrated with analysis problems where it didn't use DIRECTLY the things we spent so much time proving in class.

As an example, let's look at a standard real analysis problem:

(1/23)

The problem looks like it can be done using the dominated convergence theorem.

This is an important theorem and is used in a lot of places outside a grad real analysis class. A student would go through a lot of buildup leading to this theorem.

(2/23)

Now, let's see whether the conditions for DCT are satisfied.

First, for fixed 00 as n->infinity.

So we have a.e. convergence of the integrand in [0,1].

To the student, this builds a stronger case to use DCT to solve the problem.

(3/23)

Finding an integrable dominating function is where the problem starts.

It would seem NATURAL (at least for me) to bound it with:

p.s. it feels "natural" to me because it is reminiscent of squeeze theorem problems, math students encounter in calculus.

(4/23)

However, 1/x^n is not integrable in (0,1) and hence cannot be (further) dominated by an integrable function.

So, what I would do is since I know it converges a.e. to 1, maybe it converges uniformly to 1?

This leads me into a BAD detour.

(5/23)

How to mentally compute the cube root of the cube of a two digit number.

Thread (1/10)

In base 10, the last digits of the cubes of digits are distinct.

If d = 0, 1, 4, 5, 6, or 9, then d^3 ends in d.

If d = 2, 3, 7, or 8, then d^3 ends in 10-d.

(2/10)

So if you're given the cube of an integer, you can figure out the last digit of the cube root.

For example, if 50653 is the cube of an integer, it's the cube of a number ending in 7.

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Let n = 10a + b be a two-digit number, with a and b single digit numbers.

Then 10a <= n <= 10(a+1), and so n^3 is between (10a)^3 and (10(a+1))^3, i.e. between 1000 a^3 and 1000 (a+1)^3.

(4/10)

So suppose x is the cube of a two-digit number n = 10a + b.

Chop off the last three digits of x.

Then a, the first digit of n, is the largest number whose cube is no greater than what's left of x.

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@GWOMaths One might say that physicists study the symmetry of nature, while mathematicians study the nature of symmetry.

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Observing symmetry in nature, such as noting the similarity between the symmetries of a snowflake and a hexagon, is readily comprehensible. What does it mean then to study "the nature of symmetry"?

2/

Mathematicians define a "group", G, as a set of elements {a,b,c, ...} with a binary operation ⊡ and a distinguished element e (the identity of G) satisfying these specific properties:

3/

For all a, b, c in G
1) Closure: a⊡b is in G.
2) Associativity: (a⊡b)⊡c = a⊡(b⊡c)
3) Identity: e⊡a = a⊡e = a.
4) Inverse: There exists an element a* such that a*⊡a = a⊡a* = e.

4/

Just as functions on the integers, rationals, or reals are defined as mappings, mathematicians define a *group morphism* μ as a mapping from one group to another that preserves the group structure:

5/

Common Core math was the pinnacle of their evil genius, and low-point of our complacency... It disrupted the minds of an entire generation on how even the most basic things work.

Readin', Writin', & Rithmatic

How far did they get? Let's see.

1895 8th Grade Final Exam, Kansas:

@DesmondSwayne Sir Desmond is right. SAGE makes mathematical predictions based on flawed models.
I chose to stick my neck out almost 4mo ago here:
https://t.co/b0rT5Lq9HI
I experienced much anxiety doing this, aware that, if I was badly wrong here, I would be destroyed as a contributor.

That said, I thought the immunology so clear that I made three, specific & testable predictions which should apply if I was right.
Please examine them, on the right, under the pie chart:

In particular, I predicted that the epidemic would not reignite in London. Obviously, we have many “cases” but I urge people to disregard these & instead to look at excess deaths, on the principle that a severe (occasionally lethal) respiratory virus would show itself this way.

Here is a chart showing the trend of deaths in London alone over recent years. A prominent peak of deaths is evident as the virus swept through in spring. Also visible the August heatwave. But there are no excess deaths now.
I submit this is a striking example of herd immunity.

I’m still waiting for SAGE or indeed anyone to provide an alternative explanation for the sharp nature of the spring excess deaths peak which doesn’t involve herd immunity or otherwise removing from the pool those vulnerable to lethal outcomes (same result). Given this sharp...

The main problem with "Lift All Voices" as a justification for the Black National Anthem is the word "All." In order to achieve "All," the Woke are going to have to cook up more different unique national anthems than they have gender pronouns.

Mathematically speaking.

1/

And why would the Black community be the first to get their own anthem over the American Indians? The Indians used to have all the land, and they still do have some of it.

(use of "American Indian" here is intentional, as that's what they prefer to call themselves)

2/

This "inclusion by division" game is just the "separate but equal" logic for 1950s segregation reskinned, but when the Woke self segregate, they self segregate based on Krenshaw's Intersectional Matrix of Oppression, which in Theory is necessarily near infinite in size.

3/

When the Woke did this with gender pronouns we went from 2 genders to 37 genders to infinite genders in the span of about 5 years, but the trans folks discovered this really didn't help them out on Tinder, so they pivoted back to "trans women are women" to try and get dates.

4/

The only people left doing the "gender pronouns" game are the ones signaling their Wokeness on Twitter to their in-group, the trans women in the dating community aren't even doing it anymore. They've gone back to 2.

5/

1st grade mathematics work sheets and my mother's mathematics teaching style. Mathematics will not be so terrible, since it seems that parents are interested in preparing their little ones for mathematics before school age.

I grew up without understanding how people talk about mathematics as difficult as they do, it was my best topic at school.

IS HISTORY ALL TRUE SERIES: when even the #Calendar isn’t❓

THE REAL ACTUAL SOLAR CALENDAR:
*Based on the noble science of #Astronomy; the sister science to #Geometry, #Mathematics and the “almost” lost numerical science mysteries. Which was mastered in

“ancient” times, but is close to forgotten, in these “modern” days. Go figure❗️😄☦️

Navigation of the oceans was man’s first real great step of personal evolution, here on earth. The earliest of discovered sciences; #astronomy; allowed men to navigate through

routes to known territories; or to discover new ones, by the endlessly reliable positions of the stars and constellations, in the night sky. Hence; much diligent study, over who knows how many epochs, and great civilizations; was an esteemed scholarly pursuit. And led to a vast

understanding of the mechanics of our solar system, as well as the natural cycles of our planet, that even we; in this so-called Information Age “high civilization, don’t often know even exists any longer. A decay of common knowledge, in our “modern” educations, actually.

#Calendars have come and gone In the last few thousand years; motivated apparently by the politics of the day, or personal whim even, by the rulers of the moment; which somehow stuck, from there, and has hardly ever been questioned again.

The problems of 3rd grade mathematics work sheets are experienced by many children and their parents are usually frustrated. There is no more, there is information about how to solve the problem of your Mathematics task for children ... My suggestions will help.

Your child needs help with his homework and is useful if you as a parent know how to handle the problems of 3RD mathematics worksheets as follows:

Algebra 1 and pre-algebra worksheets are something that all would love to have been able to dominate in high school. Sometimes they do not dominate it there and that's when it could be a bit difficult.

But after learning and doing problems of algebra and pre-Algebra and worksheets online, it will no longer be a problem. There are many cases in all the life of a person where they have to learn.

Ten components of being compassionate to yourself:

1. Self-friendship: treat yourself at least as well as you would treat a friend who is going through the same situation

[thread]

2. Worthiness: remember that you have value (as all people do), and that your happiness and preferences matter.

3. Self-talk: notice when you are saying something mean to yourself (aloud, or in your head), and rephrase it in a gentler, kinder, more considerate way.

4. Connectedness: reflect on the fact that, whatever challenge you are facing, there are others who are also facing a similar challenge. You are not alone in your pain.

5. Change: note that everything changes with time. If you are dealing with feelings that are difficult to stand at this moment, they will subside. This too shall pass. Things are usually not as important or permanent as they seem to us in the moment that we are focused on them.

6. Self-empathy: orient towards yourself with tenderness, understanding and compassion, especially when you are suffering. Take on the perspective that you would towards a child or loved one that was hurting.