Notes

2-Structure of in real life like the governemt/chess are created and altered/improved
Often with starting from an idea or a few positions. Never know the entire structure first. It’s useless for mathematicians to be structuralist becuase they can never know the structures first, but start somewhere small. Problem with axioms

2-Am I postulating a number in the structure of the reals or the naturals. Mathematical practice.

Not all that matters are things in the systems and their relation.lots of other things matter. Properties of sets, lengths of intervals etc


~You say you can’t just say 4 without 3, or 5, or situating it in a structure. But why not include 3.5? It seems structuralism cares a lot about types of cardinality



Why not just use fewer ontology. Like don’t say structure of a circle or square whatever, we can say structure of a plane which includes circles and squares and anything we want to draw in it


Note: you can use axioms to describe structure. The natural number system has the structure that can be fully described with the PA. Initial, successor, induction. Instead of using the word 0 is a number. It’s saying 0 is a place, blah blah. So structuralism is just axiomatic foundations in disguise.

——————////-
We study sets, we can’t just replace sets with the natural numbers structure. Instead natural numbers can only be instantiated by sets. The natural numbers cannot capture all the properties of sets like inclusion, union etc

$$$$the fact that there are many instances is evidence that intrinsic properties and the internal systems matter!
As human beings, we have preferences and preferences are based on instrinsic properties. On top of reasoning, we have the passions(Hume) it’s what drives our preferences. So mathematical practice must cater to how we see the intrinsic properties of objects and questions about internal systems.
So people like using von Neumann over Zermelo-Fraenkel... and that we can use set theory ordinals to capture more properties like union/inclusion/intersection. But numerals can’t. We can’t easily understand those properties using minerals. They capture more things like...(math stack) than if we just think of numbers as places in the natural numbers structure.
Similar to people when talking about bastkeball, prefer one team over the other. Prefer a method to play chess than another.
——

$$$$The fact that structualism itsef defines structures based on preference:Natural numbers can have relations like less than, etc but not inclusion.
Why place more important on relations like <, division, cardinality over questions like inclusion. Since the natural numbers are isomorphic to Zermelo-Fraenkel, why can’t we say this is the Zermelo-Fraenkel fraenkel structure?
We can’t compare it with the queen bishop example. Cause we know what smarter is it has nothing to with when capturing other queen. But does inclusing have nothing to do with <? We defined < to be ... to capture the natural number system but we can use the definition of including to also capture the natural number system .



Mathematical practice. He doesn’t say why it’s obvious that we don’t use certain exemplication over others. When we speak of the natural numbers, we don’t use for instance the even or the odds.
If it is the structure and that matters why do we have these preferences.

$$$ even though we can put any object in the places like apple in 1-place, cat in 2-place. Bleh in 3-place. We don’t do it in practice. So there must be instrinsic properties to the objects like 0, {0}, etc that we like to use them. We use sets to exemplify it. We don’t write 0, {0,{0}},{0}. Similar in the world, in a basketball team, it’s not only the relations that matter, eg it is preferable that the center player is tallest in the team. Being tall does the center job better. Being y’all gets the game played more efficiently. Preferable some are shorter/taller than others.
President should have the attribute of being conpassionate, moral
If the player is tall, the game is played better, meaningful


Mathematical practice. 2 is just 2 real and 2int and 2rational. We don’t say oh we are using 2 in .... we intuitively know 2 is in all the structures. But Shapiro says 2real is the object with properties of the relation it has in the reals. 2nat is the object with the relations with the naturals. So all the twos are not the same object. So this is incompatible . Shapiro “objects are tied to the structures that constitute them” and each structure is unique
In application, we don’t talk about math in a structure we talk about things like distanctance etc .


Something about ben in this week perusall
Write position instead of placefc
Not everything is structure