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Edit - better phrasing/summary:

Maybe this phrasing helps "the same object expressed in different ways". That's one meaning behind 'equals'. 10 = 1+...4 --> 10 really is 1+...4. So if mathematically we can treat the same object in multiple ways via equals, and holism and Maudlin are saying: one way you cannot treat an object is just summing its separate parts to equal the whole, is there a conflict in something like the 10 = 1+...4 example, where LHS and RHS really are the same? Is the mathematical structure of equals and plus preserving holism?

--

I'm not trying to be provocative. I just wanted some perspective (good and bad) on this thought I had.

In mathematics, 1+2+3+4 = 10 means both sides of the equals sign are the exact same in any mathematical context. Or said differently, equal operations done to both sides will preserve the their equal relation. If left and right have some kind of difference, it is not picked out in the math. For any operation, each side will have the same mathematical behavior. In the realm of mathematics, = means each side can be swapped for all mathematical purposes. And I think even further, each side really is the other. 10 really is 1+2+3+4 and really is 2*5, etc. Just like an equation can be thought of as a function, it also has a geometric interpretation too. Neither is more correct than the other.

But for holism, we often see it summarized as "the whole is greater than the sum of the parts", a paraphrasing of Aristotle I believe. And a quote by Tim Maudlin "The world is not just a set of separately existing localized objects, externally related only by space and time. Something deeper, and more mysterious, knits together the fabric of the world. We have only just come to the moment in the development of physics that can begin to contemplate what that might be." from Interpreting Bodies.

Don't I have a plain as day conflict when I say 1+2+3+4 = 10 in the perspective of holism? Maybe mathematically they are no different, but in the physical world holism says there is some kind of difference between the parts and the whole. It seems like no physical objects can be split into any constituent objects in regards to holism, yet splitting is a mathematical operation (- and +). Objects that live in the physical world, as Maudlin says in his quote can't be so easily split. Whatever the universe is, it cannot be gotten to by thinking about individual components of it.

And could Euclid when he said "the whole is greater than the part" (Euclidean property 5 I believe), been getting at this idea? Was he avoiding saying "the whole is greater than the sum of the parts" deliberately, putting it in a milder form? Because mathematics does not make the distinction holism (of the physical world) does.

Those restrictions do not seem to enter into math. I can take any whole object, say a circle, and mathematically say it really is certain numbers added together.

Are =, +, etc possibly disconnected from the physical world?

*If this is all too vague please let me know. I don't have a great deal of mathematical philosophical knowledge but I hope I made a point.

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  • 1
    Equivalence relations expressed by = do not mean that the sides are "exactly the same", they only mean that some features relevant in the model's context match and the rest are disregarded. This is entirely neutral as far as holism/reductionism, if a model does not take enough features into account just refine it. Math supplies plenty of vehicles for that: if simple union of parts (+) is not enough put extra structure on it, specify additional relations between the elements, for example. But something always has to be disregarded because we are beings with limited resources and capacities.
    – Conifold
    Jul 13 at 19:45
  • IMO, you are reading Maudlin's "holism" in a wrong way. 10 eggs are 10 eggs also in the "physical world" and if we split the 10 eggs in four baskets, we will have always 10 eggs... provided that we do not break some of it, in which case "the whole is lesser than the sum of the parts". Jul 14 at 8:09
  • @Conifold Aren't they the same for all intents and purposes though? Like show me an a function where 1+2+3+4 vs 10 as inputs makes a difference. I thought I am basically saying 1/2 = 2/4 = 4/8. For all Mathematical operations, they behave the same. And sure you can add structure to specify additional relations, but what if we need to subtract some structure math is imposing, that isn't there physically?
    – J Kusin
    Jul 14 at 16:33
  • @MauroALLEGRANZA Then what is Maudlin's point if he isn't saying that? What you've said is very plainly what most people would think defaulty. I don't think Maudlin wrote his essay just to give the plain, default view. I think he really is saying, at least some physical objects can't be divided like you have shown. Maybe not eggs, but some physical objects.
    – J Kusin
    Jul 14 at 16:36
  • 2
    The problem is that you are reducing mathematics to = and +, or unstructured sets and their unions. Even reductionism needs more than arithmetic. Modern algebra deals with objects and operations much more complex and structured than numbers, sets and their sums and unions. "All mathematical operations" do not behave this way at all. You cannot simply splice together two groups or rings, the "sum of separate parts" does not determine the structure of the whole, operations on it. It is so even with splicing together particles in mechanics, interactions are specified independently.
    – Conifold
    Jul 14 at 19:27
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Long comment

When in mathematics we write 1+2+3+4=10 we are not making some sort of "metaphysical claim": we are asserting that when we evaluate the left-hand side expression (we "compute" 1+2+3+4) the process will terminate after a finite number of steps and the resulting value of the process will be the same as the right-hand side.

Thus, in conclusion, =, +, etc are not disconnected from the physical world, at least because humans and machines performing computations are part of the physical world.

2
  • The world of mathematics isn't making a metaphysical claim. But as soon as we represent anything in the real world with +'s and ='s, etc, aren't we in danger of losing some kind of structure of the real world in exchange for mathematical structure. Even the concept of "2 apples" is apparently not 1+1 apple in regards to holism, yet mathematically 2 = 1+1.
    – J Kusin
    Jul 14 at 16:22
  • I know this seems silly, but that is what I thought Maudlin is saying.
    – J Kusin
    Jul 14 at 16:37
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It's worth emphasising that when you type out 1+2+3+4=10, this is in the first instance a sequence of symbols. A conventional interpretation of these symbols might be something like an affirmation that numbers are metaphysical objects, that equality is a kind of identity claim, and that the addition sign is the description of a function relation on the natural numbers.

Quine argued that the most natural way to understand mathematical objects is to take their theoretical first-order posits at face value, and to thus say that since mathematical practice is a necessary part of the scaffolding of the holist scientific perspective, we ought to accept that they are a part of our reality.

But there is no specific reason within holism that requires you to interpret that sequence of symbols in that way. Indeed, according to one view often bundled with Maudlin's loose collection of ideas, the ontology of mathematical language is that of the structures of mathematically interpretable systems. It's not so much that there are actually unique objects in reality that we take to be "The numbers" - just that there are systems (of physical or otherwise scientifically important kinds) that are organized in the way described by the axioms of, say, Peano Arithmetic.

If such a practice were to provide both simpler parsimony in the ontology of the physical world and still allow us to make sense of the language and applicability of mathematics, then isn't that a preferable quality for our science as a whole? Occam's Razor, after all, invites us to question the unnecessarily duplication of entities, and to thus better evaluate our frameworks of posits taken as wholes.

1
  • I'm not trying to get into the metaphysical standing of numbers and mathematical symbols. It is the structure of the symbols and math that I am curious maps onto the structure of holism. Yes mathematical language helps us makes sense of the physical world, and is highly effective, but how is there any interpretation of its language and symbols, conventional or unconventional, that interprets +'s in a way the makes sense to holism, which is antithetical to parts and summing of parts?
    – J Kusin
    Jul 14 at 16:29
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The things equated in 1+2+3+4=10 are not the same. They are two approaches to an aspect of reality. One approach is holistic approach, saying things cannot be broken up, and the other is the reductionistic approach, saying they can. The two are incompatible or incommensurable but they can be equated.

They are inherently different though. For one people only the LHS matters, for another only the RHS matters. But the two people can talk about their differences and maybe even pull the other to the other side. If everybody would be on one side the equation would be useless.

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  • But mathematically they behave the same. There is no function that gives a different output if I plug in 10 vs 1+2+3+4. So mathematically, they are equal. Mathematically, there is no difference. And yet there is some kind of difference I would argue, and it seems you agree, outside of mathematics.
    – J Kusin
    Jul 14 at 16:46
  • They behave the same in what way? If you substitute them in a function? A collection of ten apples is different from four smaller collections. You can substitute four collections of apples in a function and you can insert the whole bunch. But still there is a difference.
    – user53288
    Jul 14 at 16:55
  • Yes mathematically, substituting them in a function, they are the same are they not? As they give the same output as inputs to any function.
    – J Kusin
    Jul 14 at 16:59
  • @JKusin I think the outcome will be different too. You can see the outcome as one whole (when you insert 10) or as a collection of "sub- wholes" (depending on the function). The outcome of f(10) can be different: new sub-wholes or one big whole.
    – user53288
    Jul 14 at 17:03
  • @JKusin Even the outcome of f(1+2+3+4) can be a new whole. If f(x)=2x then the outcome can be 2+4+6+8 or 20. I wanna upvote but I cant.
    – user53288
    Jul 14 at 17:09
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As I see it, the whole point of holism is to understand that real world objects are indeed more than the sum of their parts, something that arithmetic typically overlooks. If I were to cut you into slices (and for the avoidance of doubt I have no intention of doing so) then the parts would add up to your original body weight, although your performance as a philosopher would be significantly impaired. Arithmetic doesn’t capture this because it’s not the purpose of arithmetic to do so. I use the term arithmetic because the science or mathematics could be extended to include many if not all philosophical constructs.

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  • Thank you. And do you see mathematics in its current form as able to capture the structure of holism/holistic objects; that is, it's "loose enough", and not imposing an overriding structure?
    – J Kusin
    Jul 14 at 21:28
  • @J Kusin Certainly in a limited case one could mathematically represent holistic effects. Consider a book - we could look at its dimensions and mass, but this would tell us nothing about the information in the book. However we could be a bit smarter and count the words, sentences, and perhaps give a score of the grammatical and factual merits of the content.
    – Frog
    Jul 15 at 0:32
  • And do you think this process of being a bit smarter mathematically can continue till we hold the entire universal object (or whatever there is if there is a finality) in our grasp? This kind of serial addition. Is that counter to holism?
    – J Kusin
    Jul 15 at 0:55
  • @J Kusin maybe not - in computing we know that we can solve any 'solvable' problem (this has a particular meaning in this context), but there are insolvable problems that can not be solved. Similarly one might expect that there are problems that can not be solved mathematically, and certainly there are problems that we can't solve because we don't have the information that would be needed to do so.
    – Frog
    Jul 15 at 2:00
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I found a quote that seems to answer my question in the negative.

"Resnik maintains that holistic considerations may license the introduction of new entities, new methods, and new fundamental laws into a theory, but that they do not extend to the more global theories, where mathematics is singled out as the most global theory we possess." From The Philosophy of Mathematics Today ed. Matthias Schirn

So mathematics is able to encompass holism, as it is the most global theory we posses.

I'm sure there are other takes. I think one can be seen later in the work:

"...but with mathematics being the most global theory an a priori-empirical distinction threatens to resurface. In the final section of the chapter I will attempt to deal with this worry."

I take this to mean, which comes first, a priori reasoning or a posteriori empirical results? Can we really a priori say mathematics can handle any empirical result?

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