Suppose we take “applied mathematics” in an extremely broad sense that includes math developed for use in electrical engineering, population biology, epidemiology, chemistry, and many other fields. Suppose we look for mathematical structures that repeatedly appear in these diverse contexts — especially structures that aren’t familiar to pure mathematicians. What do we find? The answers may give us some clues about how to improve the foundations of mathematics!

You can see my talk slides here. You can click on any picture or anything written in blue in these slides to get more information — for example, references.

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14 Responses to The Foundations of Applied Mathematics

What about software engineering:-? It uses e.g. UML (Unified Modeling Language), a suite of graphical notations which can communicate a number of different static and dynamic views of systems, especially software-intensive systems.

I’ve heard about UML, but haven’t thought about it lately – thanks for reminding me! We talked about it before on this blog, and I decided that it was a bit of a ‘grab-bag’ of tools rather than a simple mathematically elegant thing… so I decided to start by focusing on the mathematical theory of some other kinds of networks: stochastic Petri nets, electrical circuits, bond graphs and signal-flow graphs. But I want to keep expanding my scope, so I should get up my courage to understand UML at some point… and Systems Biology Graphical Notation, as I mention in my talk.

Yes my experience as a sporadic user of UML tends to confirm the grab-bag characterization.

That said, there seems to be an attempt to define at least some of the diagram types with care. Kinda-sorta consistent w/ this assertion, a) I gather that there are a number of UML-consuming SW tools out there, b) there is a UML metamodel and meta-metamodel.

Also see David Harel’s statecharts, which do have formal semantics and are now part of the UML cornucopia. Harel is also advocating for statecharts in biology, and has shown quantitative models for example in cell development with nice simulation results.

Also see David Harel’s statecharts, which do have formal semantics and are now part of the UML cornucopia.

We had a discussion about UML, statecharts and Petri nets here and here. I was not ready, at the time, to put in the work to understand what statecharts are! I may still not be ready. So maybe you can get more out of this discussion than I did.

arch1 wrote:

(to which I can’t directly attach this response as the wiki tool gives me no Reply option)

Comments on comments on comments get skinnier and skinnier and skinnier, so at some point this process cuts off, and then you go back up the comment tree to the first point a reply is allowed. You did fine!

It’s “how can you step in the same river twice?” And the answer is: you don’t, you step in two isomorphic rivers. The isomorphism is the passage of time.

I only knew that as a saying: “You never step in the same river twice”.

Regarding the mystery of time, here’s a perhaps more puzzling paradox of Nagarjuna: If cause precedes effect, how does it bridge time? If cause and effect were simultaneous, how to discern cause and effect? (From autocommentary to verse 6 of sunyatasaptati. Hume has another argument that cause and effect can’t be simultaneous.)

I’d really like to understand why you placed the emphasis on “reversible” in slide #26:

There is some reversible process f taking us from x to y.

What if the process is instead irreversible like many real physical processes ?

The way i see it, Petri nets, like FSM Diagrams, are Process Diagrams (that describe how the state of the system evolves over time), while signal flow graphs are Activity Flow Diagrams. The latter type are in fact often used to express computation processes.

I am not sure if there are clear boundaries though, but perhaps a table categorizing the known diagrams as belonging to one (or more ?) of the three types could help the audience … just a thought.

Btw, I would have loved to see the talk, too bad i’ll be traveling this weekend ….

What if the process is instead irreversible like many real physical processes ?

We’re a bit less likely to treat and as equal if we’ve got a morphism that’s not invertible. However, it’s not a hard-and-fast rule.

Examples from physics are a bit confusing for me here.My favorite counterexample comes from theory of rewriting—which you may understand well, because it’s important in symbolic computation.

In rewriting theory we often want a set of rules for rewriting expressions that is terminating (the process always ends) and confluent (even if we have different choices about how to rewrite an expression, we can always make choices later on that lead to the same final result):

This guarantees that if we keep using our rules to rewrite an expression, the process eventually leads to a unique final answer.

Rules that are terminating and confluent can never be reversible! Nonetheless, if we type something like

into Matlab and it spits out

it has probably used terminating and confluent rules to do this calculation… and mathematicians are willing to summarize this with an equation:

I think the notion of scientific truth (as opposed to logical truth) itself can be modeled through category theory? As in some maps of the world are more truthful than others because they better preserve some kind of relationships (which can be quantified, as in a metric) in the world?

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You need the word 'latex' right after the first dollar sign, and it needs a space after it. Double dollar signs don't work, and other limitations apply, some described here. You can't preview comments here, but I'm happy to fix errors.

What about

softwareengineering:-? It uses e.g. UML (Unified Modeling Language), a suite of graphical notations which can communicate a number of different static and dynamic views of systems, especially software-intensive systems.I’ve heard about UML, but haven’t thought about it lately – thanks for reminding me! We talked about it before on this blog, and I decided that it was a bit of a ‘grab-bag’ of tools rather than a simple mathematically elegant thing… so I decided to start by focusing on the mathematical theory of some other kinds of networks: stochastic Petri nets, electrical circuits, bond graphs and signal-flow graphs. But I want to keep expanding my scope, so I should get up my courage to understand UML at some point… and Systems Biology Graphical Notation, as I mention in my talk.

Yes my experience as a sporadic user of UML tends to confirm the grab-bag characterization.

That said, there seems to be an attempt to define at least some of the diagram types with care. Kinda-sorta consistent w/ this assertion, a) I gather that there are a number of UML-consuming SW tools out there, b) there is a UML metamodel and meta-metamodel.

UML also differs in that most of it is not quantitative.

Also see David Harel’s statecharts, which do have formal semantics and are now part of the UML cornucopia. Harel is also advocating for statecharts in biology, and has shown quantitative models for example in cell development with nice simulation results.

Good point Arrow (to which I can’t directly attach this response as the wiki tool gives me no Reply option)

(BTW, I just noticed – 1,000,039 hits:-)

Jon Rowlands wrote:

We had a discussion about UML, statecharts and Petri nets here and here. I was not ready, at the time, to put in the work to understand what statecharts are! I may still not be ready. So maybe you can get more out of this discussion than I did.

arch1 wrote:

Comments on comments on comments get skinnier and skinnier and skinnier, so at some point this process cuts off, and then you go back up the comment tree to the first point a reply is allowed. You did fine!

Cool!

What is that “old puzzle of Heraclitus”?

It’s “how can you step in the same river twice?” And the answer is: you don’t, you step in two isomorphic rivers. The isomorphism is the passage of time.

I only knew that as a saying: “You never step in the same river twice”.

Regarding the mystery of time, here’s a perhaps more puzzling paradox of Nagarjuna: If cause precedes effect, how does it bridge time? If cause and effect were simultaneous, how to discern cause and effect? (From autocommentary to verse 6 of sunyatasaptati. Hume has another argument that cause and effect can’t be simultaneous.)

Ergo, time must be discrete.

I’d really like to understand why you placed the emphasis on “reversible” in slide #26:

What if the process is instead irreversible like many real physical processes ?

The way i see it, Petri nets, like FSM Diagrams, are Process Diagrams (that describe how the state of the system evolves over time), while signal flow graphs are Activity Flow Diagrams. The latter type are in fact often used to express computation processes.

I am not sure if there are clear boundaries though, but perhaps a table categorizing the known diagrams as belonging to one (or more ?) of the three types could help the audience … just a thought.

Btw, I would have loved to see the talk, too bad i’ll be traveling this weekend ….

Giampiero wrote:

We’re a bit less likely to treat and as equal if we’ve got a morphism that’s not invertible. However, it’s not a hard-and-fast rule.

Examples from physics are a bit confusing for me here.My favorite counterexample comes from theory of rewriting—which you may understand well, because it’s important in symbolic computation.

In rewriting theory we often want a set of rules for rewriting expressions that is

terminating(the process always ends) andconfluent(even if we have different choices about how to rewrite an expression, we can always make choices later on that lead to the same final result):This guarantees that if we keep using our rules to rewrite an expression, the process eventually leads to a unique final answer.

Rules that are terminating and confluent can never be reversible! Nonetheless, if we type something like

into Matlab and it spits out

it has probably used terminating and confluent rules to do this calculation… and mathematicians are willing to summarize this with an equation:

Sir can you give some pointers for category theory as i am new to this topic.

I think the notion of scientific truth (as opposed to logical truth) itself can be modeled through category theory? As in some maps of the world are more truthful than others because they better preserve some kind of relationships (which can be quantified, as in a metric) in the world?