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

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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"?

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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:

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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.

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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:

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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

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:

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.