The beauty of category theory(!) is that $5 * $3 CAN make sense: - In fact, I came up with a way of multiplying in the dollar category A: $5 by $3 exactly the same way as you would in the quantity category B: 5 boxes by 20 widgets. These can be seen as isomorphic transformations, each resulting in a categorical product. How? First, they both have the SAME kind of constraint: You need to know how one quantity translates into the other. Moreover, there IS a functor between the two transformations going from B to A. In category B, you would get 5 boxes x 20 widgets = 100 ONLY if EACH box is the same size and a full box contains 20 widgets. However, the real result is actually a product (5 boxes, 100 widgets) not a just a number or scalar. The same way, for US$ 5 x CAN$ 3 to work in category A, you need to know how much Canadian dollar goes into 1 US dollar. But you can get US$5 x CAN$3 even if US$1 is worth more or less than CAN$ 3. That's the beauty of this categorical product. Similarly, if one full box contains less or more than 20 widgets, 5 boxes x 20 widgets no longer equals 100. But as a product it still makes sense. Suppose now that we set the constraint for category B that each box must go to a different address and 1 box fits 25 widgets, meaning with 20 a box will be 0.2 empty and only 0.8 full. Then, if you set the same rule of equal distribution into each box as you have equal exchange into each unit for currencies, you get 5 boxes x 20 widgets = (5x0.8b, 100w) as a product. You can translate this to (4b, 100w) ONLY of you are happy to leave 1 box empty, ship only 4 boxes and make one of the 5 recipients unhappy because they receive nothing of the widgets. More generally and thinking categorically, the T: $ x $ product is a discreet matrix or continuous field (consider a fractional exchange rate) of monetary assets along two different dimensions. They can be two arbitrary but convertible currencies, or two different persons using the same currency, etc. Thus a Category where objects are $^2 can describe a series of valuations, exchanges and transactions not just by adding or multiplying like 3 x $5 or $3 = $5 as in the accounting books referred by David at 10:00, but along two different dimensions describing a network of transactions that may even include loops or other basins as in some regularly occurring circular exchange, for example.
The ACT-SME dialogues reminded me of Cognitive Behavioral Therapy, and I think the likeness makes total sense. The SME is the patient, their subject being their own life, mind, and behaviors. The ACT is the therapist and doesn't substantially know the subject but understands the fundamental patterns. The therapist must reopen closed questions about the subject to examine its structure so that they can figure out ways for the patient to deal with it in a healthier, more manageable way.
I can relate to the plumbing analogy. Another analogy I like is the idea that the industrial revolution was drive more by bookkeeping than the steam engine.
This is precisely what (good) software engineers do the best, they build the right abstraction by talking to SMEs. They just use other modeling languages than Category Theory.
The first incarnation of this talk was given at the NIST ACT workshop in early November 2022. I'll link to other reincarnation of NIST ACT talks here. Pawel Sobocinski: th-cam.com/video/Yh_ASMEODVs/w-d-xo.html
Great talk. Seems like an excellent video for the a Category Theorist to summarize their role to the other folks in the room/project in a concise and relatable way.
To give you some language to help build your case, when you talked about AT&T, the best broadly understood word I've found is "network effect" which moves along a value curve as a slow-building exponential. And CATMATH models certainly exhibit that, but fortunately dont necessitate it to such a vicious degree. - Doug P.
I think for some recipes, due to like, how long different reactions take, where you need to proceed to the next step before something happens too much, some steps may *need* to be done in parallel. I don’t know if that happens much in practice.
@@markhathaway9456 hm, I think computer programs can still be run without parallelism by just simulating the parallelism , at the cost of being slower? I’m saying that I think for some recipes, this might not work in that the food which is the final result, might not come out right. If “getting the output fast enough” is considered part of the “result” of the computation though, then sure.
I have a below zero level in math but somehow i still fancy losing hours listening to talks about this topic like it’s poetry or something
The beauty of category theory(!) is that $5 * $3 CAN make sense: - In fact, I came up with a way of multiplying in the dollar category A: $5 by $3 exactly the same way as you would in the quantity category B: 5 boxes by 20 widgets. These can be seen as isomorphic transformations, each resulting in a categorical product. How? First, they both have the SAME kind of constraint: You need to know how one quantity translates into the other. Moreover, there IS a functor between the two transformations going from B to A.
In category B, you would get 5 boxes x 20 widgets = 100 ONLY if EACH box is the same size and a full box contains 20 widgets. However, the real result is actually a product (5 boxes, 100 widgets) not a just a number or scalar. The same way, for US$ 5 x CAN$ 3 to work in category A, you need to know how much Canadian dollar goes into 1 US dollar. But you can get US$5 x CAN$3 even if US$1 is worth more or less than CAN$ 3. That's the beauty of this categorical product. Similarly, if one full box contains less or more than 20 widgets, 5 boxes x 20 widgets no longer equals 100. But as a product it still makes sense.
Suppose now that we set the constraint for category B that each box must go to a different address and 1 box fits 25 widgets, meaning with 20 a box will be 0.2 empty and only 0.8 full. Then, if you set the same rule of equal distribution into each box as you have equal exchange into each unit for currencies, you get 5 boxes x 20 widgets = (5x0.8b, 100w) as a product. You can translate this to (4b, 100w) ONLY of you are happy to leave 1 box empty, ship only 4 boxes and make one of the 5 recipients unhappy because they receive nothing of the widgets.
More generally and thinking categorically, the T: $ x $ product is a discreet matrix or continuous field (consider a fractional exchange rate) of monetary assets along two different dimensions. They can be two arbitrary but convertible currencies, or two different persons using the same currency, etc. Thus a Category where objects are $^2 can describe a series of valuations, exchanges and transactions not just by adding or multiplying like 3 x $5 or $3 = $5 as in the accounting books referred by David at 10:00, but along two different dimensions describing a network of transactions that may even include loops or other basins as in some regularly occurring circular exchange, for example.
The ACT-SME dialogues reminded me of Cognitive Behavioral Therapy, and I think the likeness makes total sense. The SME is the patient, their subject being their own life, mind, and behaviors. The ACT is the therapist and doesn't substantially know the subject but understands the fundamental patterns. The therapist must reopen closed questions about the subject to examine its structure so that they can figure out ways for the patient to deal with it in a healthier, more manageable way.
I can relate to the plumbing analogy.
Another analogy I like is the idea that the industrial revolution was drive more by bookkeeping than the steam engine.
This is precisely what (good) software engineers do the best, they build the right abstraction by talking to SMEs.
They just use other modeling languages than Category Theory.
The first incarnation of this talk was given at the NIST ACT workshop in early November 2022. I'll link to other reincarnation of NIST ACT talks here.
Pawel Sobocinski: th-cam.com/video/Yh_ASMEODVs/w-d-xo.html
Great talk. Seems like an excellent video for the a Category Theorist to summarize their role to the other folks in the room/project in a concise and relatable way.
To give you some language to help build your case, when you talked about AT&T, the best broadly understood word I've found is "network effect" which moves along a value curve as a slow-building exponential. And CATMATH models certainly exhibit that, but fortunately dont necessitate it to such a vicious degree.
- Doug P.
Absolutely brilliant video!
This advice is so helpful
I think for some recipes, due to like, how long different reactions take, where you need to proceed to the next step before something happens too much, some steps may *need* to be done in parallel.
I don’t know if that happens much in practice.
In computer programming it's more and more common. Probably most of the top languages of today offer concurrency (parallelism) in one form or another.
@@markhathaway9456 hm, I think computer programs can still be run without parallelism by just simulating the parallelism , at the cost of being slower? I’m saying that I think for some recipes, this might not work in that the food which is the final result, might not come out right.
If “getting the output fast enough” is considered part of the “result” of the computation though, then sure.
Category Theory strikes again! ^.^
work
Another monetery unit dimension is time I think.