(𝑎 + 𝑏)² = 𝑎² + 2𝑎𝑏 + 𝑏²

... but fried more deeply ! 😄

#maths

Does anyone know how to change the caret from a thin almost invisible vertical line in ghostty to a box like the one in macos Terminal?

#ghostty_terminal #ghostty #terminal #macos

techie question:

Is it better to format removable flash storage as exFAT and not Apple's APFS?

Use case is not file transfer between systems but local backup.

#tech #filesystems #macos #apple

Looking forward to doing a talk on the Grammar of Graphics with python at the next PyData Cornwall meetup.

It's an opinion, but I do feel python's uber-popular matplotlib has very low conceptual coherence, and is far from declarative.

https://www.meetup.com/pydata-cornwall/events/317063147/

#pyhton #cornwall #pydata #datascience #grammarofgraphics

The Cambridge Python User Group meetup is apparently shutting down.

:(

#python #cambridge

whats that word to describe a sequence of things that have no gaps between the elements

eg

5, 6, 7, 8

but not

5, 6, 8, 9

my brain has frozen and all I can think of is "adjacent" but that isn't it

#maths

Current exercise - prove strong induction using simple (piano axiom) induction.

Very cool if I can do it!

#maths

Am i the only one who thinks Strong Induction is mis-named?

The assumptions are stronger so the principle is weaker - imho.

What am I missing?

#maths

started making slow but sure progress on working through Analysis I by Tao.

4 exercises has taken me 2 weeks!

https://tao-analysis-solutions.blogspot.com

#maths

Is this a correct use of the contrapositive?

I'm uncertain as the contrapositive has a false antecedent and a false consequent?

for ref: axiom 2.3 says the successor on any natural number is never 0, and axiom 2.4 says x++=y++ implies x=y

#maths #logic

Imagine if there was (were?) a "subjunctive mood" in mathematics.

I'd use it in "assume for the purpose of contradiction" parts of a proof.

So instead of "x is 3" we'd say "x would be 3".

#mathematics #maths

I was doing an exercise to prove an <=>

And the usual way is to prove both => and <=

But the paths were different - the asymmetry bothers me.

Is there something to be learned about this asymmetry of reasoning (application of axioms and low-level lemmas) ?

#maths

If 𝑎++ exists, does that man 𝑎 exists?

Here we are working only with natural numbers, and at the level of the Peano axioms.

---

Why am I asking? In an induction proof, the induction hypothesis is of the form ..

"if 𝑎++ is a natural number then X Y Z"

And I want to know if I can then assume 𝑎 exists and use it.

#maths

I'm writing up solution exercises.

Which is clearer (for a more beginner audience) ?

A - more narrative

B - less "wall of text"

1/2

#maths #education #selfteaching

What does this sticker mean?

Does it mean I can't take the sticker off?

It's on a new TV.

#tech #tv

is there a macos gui tool for updating my codeberg repository?

my use case is super simple - i edit files and upload/sync them to codeberg.

i'm currently manually uplaoding the files via the web page.

i'd love to click a button to do it all.

#codeberg #macos

If

𝑥 = 𝑦 + 𝑧

and

𝑦 = 𝑎 + 𝑏

If we're only using Peano axioms and the axioms of equality (picture attached), how can we justify the following?

𝑥 = (𝑎 + 𝑏) + 𝑧

It's not the temptingly named subtitution axiom. I don't think ...

#maths

I've finished 🎉 all the exercises* in Number Theory Step by Step ... this self-teaching project has taken me just days less than a year!

I've written up my solutions - I've learned a lot by doing them, and yes I may have got some wrong, or misunderstood some ideas, but overall I have progressed far beyond where I was last year - especially on a topic my brain has avoided for 40 years.

https://numbertheorystepbystep.blogspot.com

---

* except the supplementary ones

Next steps - work through and do the exercises in Analysis I by Tao.

#maths #numbertheory #selfteaching

What property is required of 𝑓(𝑥) for the following to be true?

𝑓(𝑎) < 𝑓(𝑏) ⇔ 𝑎 < 𝑏

Is it simply that 𝑓(𝑥) is a monotonically increasing function? That is, the gradient is non-negative.

---

And a follow up.

If the domain of 𝑓 is 𝑥≥1 does the gradient need to be non-negative for 𝑥>1 or 𝑥≥1 ?

#maths #analysis

A map of things you'll learn in this beginner's intro to Prolog.

free content:
https://prologbyexample.blogspot.com/p/toc.html

book form:
https://www.amazon.com/dp/B0BTQ7P69H/

#prolog #gofai #AI