MF42b: Deflating modern mathematics: the problem with `functions'

[Second of two parts] We address a core logical problem with modern mathematics--the usual definition of a `function' does not contain precise enough bounds on the nature of the rules or procedures (or computer programs) allowed.

Here we discuss the difficulty in the context of functions from natural numbers to natural numbers, giving lots of explicit examples. WARNING: this video and the last one destabilizes much of the mathematics taught in universities.

This video belongs to Wildberger's MathFoundations series, which sets out a coherent and logical framework for modern mathematics.

Category:

Science & Technology

Tags:

• Mathematics
• foundations
• function
• problem
• rule
• procedure
• Wildberger
• definition
• part
• two

[neuraxon77]

askNorm(42)

[njwildberger]

It is 19.

[eedahl]

??? It's a collection of ordered pairs mapping a set X into a set Y, what's the problem? Your 'philosophical'" problems are best addressed in a discussion on foundations, or better yet, in applied mathematics. Many things are functions even though they are not effectively computed. I really wonder what your next video is.

[njwildberger]

This is a series called `MathFoundations'.

The title of the next video is found at the end of each video.

[mediteight]

I'm sorry but you really really need to at least mention the usual and precise definition of a function that mathematicians actually use. Without doing this it looks awfully like a straw man argument...

[njwildberger]

`usual and precise' don't apply equally in this case...

See my comments to your question in the last video.

[sk8hack]

what is the inverse askNorm function of 20?

[thekkl]

I have trouble understanding a lot of the math here, but let me answer a question posed in this video that comes more easy to me.

Q: 'What is the sound of one hand clapping?'

A: Against what?

You see, when people say to clap our hands, what they mean without actually saying it is to clap one of your hands against another, and they save time by just saying clap hands. To clap one hand you need to clap it against something. By definition this is an incomplete question.

[mediteight]

The usual function definition does give a perfectly well defined bound to what is meant by an acceptable rule...

The boundary on what is considered an acceptable rule or not is precisely decided by whether or not the rule defines a subset of SxT in which for each s belonging to S there exists a unique (s,t) belonging the subset.

Which part of this isn't clear (or is inconsistent in some way)?