[FRIAM] Can empirical discoveries be mathematical?

[Roger Frye]

Reuben had an article in Issue 65 of Eureka Magazine titled 'Solving
Problems by "Cheating": Operational Calculi, Function Theory, and
Differential Equations'. The article is a compilation of tricks that he ran
across during his career that seemed to apply in a general way to solving
problems. The theme is that you doodle with methods that you have no right
to assume would work in this particular case, and if you get something
worthwhile, then go back and prove it.

Towards the end of his life he became more interested in the metaphors that
are at the basis of mathematical thinking, the bodily actions that have
been abstracted into mathematical concepts. Yuri I. Manin also spoke of
Mathematics as Metaphor is a slightly different way in his essays.

-Roger


On Mon, Sep 6, 2021 at 8:34 PM Frank Wimberly <wimberly3 at gmail.com> wrote:

> Our late friend Reuben Hersh was interested in these questions.
>
> ---
> Frank C. Wimberly
> 140 Calle Ojo Feliz,
> Santa Fe, NM 87505
>
> 505 670-9918
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>
> On Mon, Sep 6, 2021, 7:58 PM Eric Charles <eric.phillip.charles at gmail.com>
> wrote:
>
>> As I said a few days ago: I think traditionally,  "mathematical" would
>> have been synonymous with "rigorous deduction from a minimal number of
>> axioms", but I doubt that approach is clear cut anymore.
>>
>> I am pretty confident that modern mathematics is WAY more open-field than
>> that.  The Stanford Encyclopedia of Philosophy seems to agree with that
>> intuition, though I think it is an even broader topic than implied by just
>> this entry:  Non-Deductive Methods in Mathematics (Stanford Encyclopedia
>> of Philosophy)
>> <https://plato.stanford.edu/entries/mathematics-nondeductive/>
>>
>>
>> <echarles at american.edu>
>>
>>
>> On Mon, Sep 6, 2021 at 11:19 AM Barry MacKichan <
>> barry.mackichan at mackichan.com> wrote:
>>
>>> Briefly, and in my opinion, mathematics can only make claims like ‘if A
>>> is true then B is true’. To say B is true, you must also say A is true.
>>> Eventually you have to go back to the beginning of the deductive chain, and
>>> the truth of the initial statement is inductive, not deductive or
>>> mathematics. You can predict the time and place of an eclipse, and this
>>> prediction is based on mathematics and a mathematical model of reality —
>>> Newton’s laws in this case. But the truth of this prediction is inductive
>>> since the initial positions and velocities for the calculation are
>>> inductive, as is the applicability of Newton’s laws to reality, and even
>>> the ‘fact’ that mathematics can describe the universe is inductive.
>>>
>>> And Einstein showed that the applicability of Newton’s laws was in fact
>>> wrong and offered a new model — which we inductively accept as true, if
>>> only provisionally.
>>>
>>> Mathematics cannot prove any statement about the real world. Any such
>>> statement will depend at some point on an inductive truth or a definition.
>>>
>>> —Barry
>>>
>>> On 3 Sep 2021, at 18:10, thompnickson2 at gmail.com wrote:
>>>
>>> Ok, is mathematics (logic, etc.) a way of arriving at true propositions
>>> distinct from observation or are mathematical truths different from
>>> empirical truths?
>>>
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