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#maths

30 posts20 participants5 posts today

I don't want to ask if the attached derivation is correct - that's a boring request.

Is there an online tool I can use to check for me?

The few tools I have found don't do second-order λ2 terms.

Advice welcome...

I just learned that in simply-typed λ-calculus, you can't type

λx . xx

but with λ2, you can, as follows

λx : (Πα : ∗. α →α) . x(σ →σ)(xσ)

---

I'm still trying to get my head around the fact that each x has different (type) arguments.

Feels like cheating. Feels like whatever is chosen for x at the λ, should be set in stone for the remainder of the scope of λ.

1/2

Sanity check needed :

✢ If a term M is legal in λ2,

✢ And if a term N is also legal in the same context,

✢ Then MN is also legal, right?

---

This seems intuitively obvious but I can't actually find a definition / lemma in my textbook to confirm it.

The Subterm Lemma seems to be the opposite of what I'm seeking. It says that if MN is legal, then the sub terms M and N are legal too.

Science models of the world are useful, but we know they're wrong (they're only models)

New Books in Physics and Chemistry: Grace Lindsay, "Models of the Mind: How Physics, Engineering and Mathematics Have Shaped Our Understanding of the Brain" (Bloomsbury, 2021)

Episode webpage: newbooksnetwork.com

Media file: traffic.megaphone.fm/NBNK90615

New Books NetworkHomepage - New Books Network

"It is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. This shows that the mathematical language has more to commend it than being the only language which we can speak; it shows that it is, in a very real sense, the correct language." – Eugene Wigner (1902-1995)
#quote #mathematics #math #maths

my brain isn't working

did I get the type on the last line right?

---

if x was x:β then the type would be β→...

but here x is x:Πα:*.(α→α) and naively that means the type is Πα:*.(α→α) →...

I'm partly worried that α should be σ but then I remember that α is bound so irrelevant outside the scope of the Π

I'm begrudgingly impressed by overleaf - an online latex editor and previewer.

They managed to include useful latex packages, have good documentation and tutorials, have a UI that isn't too complicated and headache-inducing.

I still use offline lyx for my projects, but overleaf is good for generating small snippets of typeset maths.

Begrudging? Well, I wanted to create an online version of lyx (no latex code) but never managed to find time.

overleaf.com/

www.overleaf.comOverleaf, Online LaTeX EditorAn online LaTeX editor that’s easy to use. No installation, real-time collaboration, version control, hundreds of LaTeX templates, and more.
Continued thread

Why am I asking?

previously I worked out the type of

λxyz. xz(yz)

as

(B → (A → C)) → (B → A) → B → C

where x:A, y:B, z:C

---

now I want to parameterise the types of x,y,z to make a λ2-term and it seems easier if I can do the funny types

λα,β,γ:*. λx:(β → (α→γ)) . λy:(β→α) . λz:β. xz(yz) : Παβγ:*.(β → (α→γ)) → (β→α) → β → γ

If I wasn't allowed to have non-trivial types for x, y, z then the "algebra" on the RHS would be really convoluted

---

2/2

Is it normal to do this kind of thing?

λα,β:* . x:α→β, y: α . xy : Πα,β:*. (α→β) →α →β

That is, to have non-trivial types for objects like x?

--

To make that clearer ...

So instead of just doing x:α, y:β we do things like x:α→β→β and y:β→α

---

1/2

Replied in thread

@jonmsterling

Thanks

---

I previously struggled to find a T such that

Πα:*.α : T

which appears to ask for a 'type of a type'

with some help, I used the formation rule to say T must be *

---

The reason I'm struggling a little with this kind of expression again is I have a more challenging (to me) exercise where I have to find an inhabitant for α and β given the context

α : ∗
β : ∗
x : α →Πα:*.α
f : (α →α) →α

looking at x is why I'm thinking " what is the type of Πα:*.α ?"

"In mathematics it is new ways of looking at old things which seem to be the most prolific sources of far-reaching discoveries. A particular fact may have been known for centuries, and it may have been sterile or of only minor interest all that time, when suddenly some original mind glimpses it from a new angle and perceives the gateway to an empire." – Eric Temple Bell (1883-1960)
#quote #mathematics #math #maths