calc logical interpretation of empty cells / arguments 'not intuitive'

[newbie-02]

@Lupp: whow, you got me / got it, ‘f(empty)’ needs to be the neutral element for ‘f’. thus PRODUCT(nothing)=1 is ok. sorry, i was slow.

questions left:

what is the neutral element for a logical function / operation?

is the reference to a blank cell something like ‘having no operand / argument’ or more something like ‘the argument / operand is something associateable with nothing/no worth/zero’?

while writing i get the idea you’ll know and teach me something about the distinction between functions and operations … sorry again, at this point i miss at least 40 years of ‘practising math’.

i’m quite stretched, threw your ideas in a discussion last night and was ‘thrown back’ with:

• lambda Kalkül
• recursion
• y-Kombinator
• Kategorie-Theorie
• Hilberts Hotel
• aleph Abstufungen
• Brady - youtube
• computerphile
• numberphile
• 3blue1brown

and just now ‘math logic is broken anyway’: infinite + infinite = infinite → infinite - infinite = infinite???

i’ll mail …

[Lupp]

There is a well established habit in Boolean Algebra to write disjunction (OR, better adjunction?) with the + as the operator an conjunction (AND) with *. This is corresponding with my suggestion above to specify TRUE as the value to be returned by AND(EmptySequence) and FALSE for OR(EmptySequence).
Apart from formal arguing relying on recursion ar analogies you may “convince” yourself of AND(EmptySequence)=TRUE being “intuitive” by the translation of “All the elements need to be TRUE.” to “All the elements must not be FALSE.” An empty seuence cannot contain a FALSE element after all.

[Lupp]

Concerning your slightly smiling remarks about broken math logic I need to admit that I’m not sufficiently familiar with English terminology insofar. Doing away with the smile I would state:
Treatment of “uneigentliche Elemente” and allowing for “Ausartungsfälle” are related topics, but also distinctly distinct. This in the sense that introducing “uneigentliche Elemente” some distinctions based on pragmatic judgements can more easily be accepted.
Well known example: Adding “uneigentliche Elemente” to the field of real numbers generally is done by introducing two of them: {+∞, -∞}. Having embedded the real numbers into the field of complex numbers this doesn’t make sense any longer. For many purposes you will accept only one infinite element, and subjoin {∞}. If you want to do alanlytical geometry using complex numbers, you will need an infinite set of “uneigentliche Elemente” to get a model for the classes of parallel lines. …

[Lupp]

And you may develop a non-standard theory of “calculus” treating the infinitely small elements you then need to allow for as “uneigentliche reelle Zahlen”. The set of “uneigentliche Elemente” then is of higher cardinality by an infinite order than the “eigentliche” numbers.
You may also still doubt whether it was a final decision in history of math to accept the concept of “actual infinity” which was formerly rejected by many. (“Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können.” D. Hilbert. That doesn’t sound math.) Anyway the critical problems of modern math (related to undecidability) seem to not be introduced by that decision.