Okay

welcome to SMI this week.

Are there any questions?

We're still in the phase of talking about what math does and math language is in general.

We've started talking about mathematical argumentation and proofs and I've tried to convince you that there are a couple of language rules that's all proof is.

You're arguing for the truth of certain matters and this game has certain rules.

And we've looked at these rules.

I call them proof talk.

Right

we've looked at various proofs and I've revealed to you that these rules are often also called syllogism

makes them.

Makes them feel special.

Right

so really what a syllogism is

is we have a couple of assumptions which are assumed to be true and we reach a conclusion from them.

In a proof by contradiction

if we want to prove A by contradiction

you assume non-A to be true.

You continue your proof until you get into trouble

which is exactly where we want to be.

And that allows us to conclude that A is indeed true because not A cannot be true.

We've looked at the naming rule which allows if we are convinced that a certain object exists

we've argued for that

we've justified that.

And if it exists

we can give it a name

which is very convenient because then we can call it by name.

There's the proof by local hypothesis.

If we want to prove something of the form if A

then B.

We can do that by assuming A to be true and then we construct a proof that actually usually uses A of B.

And then we can

we have that proof from the assumption A

we can conclude B.

We can use that proof, seal it off.

You're never going to look into it again because you're not allowed to use A anywhere outside of that.

And then we kind of make a bow around it

package it

and conclude A implies B.

And if we have if A

then B.

And we also have A, we can conclude B.

It's called chaining.

And very often we have things like for all X

A

then B.

And in the chaining rule, we're allowed to instantiate, i.e.

replace by actual values all these X's.

There may be more than one.

And on the other hand