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Is Mathematics a Natural Science? (Is that important?)

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In our time there is a tendency to treat Mathematics as though it is a natural science.

Black Swan with Cygnets

This reflects in part the shift in meaning of the term Science in recent centuries, from knowledge and systematic bodies of more or less established knowledge, to the natural sciences based on inductive reasoning on observation and experiment. Where, inductive here denotes arguments whereby evidence — typically empirical — supports but does not logically demonstrate a conclusion, as a rule provisionally. Such has been multiplied by Scientism, the view, assumption or implication that Science ring fences and monopolises reliable, serious knowledge. (Of course, such Scientism is self-referentially incoherent as this is an epistemological and thus philosophical claim; it fails its own test.)

In a current thread, this has led to an exchange, worth headlining for reflection:

>>KF, 4: . . . PS: I have also recently seen someone describing Mathematics as a “Science” — a sign of Scientism’s attempt to monopolise all serious knowledge. Of course Scientism is itself self-referentially incoherent. However, more importantly, Mathematics is precisely not a discipline in which theoretical constructs are empirically tested and are taken as a sort of weak form, provisional knowledge due to empirical reliability. We need sterner stuff, rooted in logic and coherence, driven in the end by self-evident first principles of right reason. For example, number itself pivots on distinct identity, e.g. A vs ~A leads to 1 and 2 etc. Indeed, this pattern of being rooted in logic is part of why Mathematics plays the role of a plumbline in considerations on scientific endeavours. We need the logic of structure and quantity (including space etc) to be a standard of reference. That we can do Mathematics is a sign.

BO’H, 5: Eh? The four colour map theorem was a theoretical construct that was empirically tested: they narrow down the possible maps, and then used a computer to literally try every combination. Other postulates are certainly “a sort of weak form, provisional knowledge due to empirical reliability”, because not every combination can be tested, and no other proofs are available (e.g. Golbach’s conjecture).

KF, 6: . . .  mathematics has an emphasis on axiomatic systems and on results derived therefrom by logical, step by step proofs, accumulating into what is now a huge body of knowledge. You know full well that proofs are generally not done by empirical examples and making an inference to generality or to the best current explanation or the like. [NB: I made an error regarding BO’H’s background, corrected later; not material.]

BO’H, 9: . . . I gave an example @5 where mathematics was done empirically. You just ignored that comment . . . [Actually, posting difficulties — Internet access here is spotty right now — and events intervened, then I thought I could wait.]

KF, 10: . . . proof of a finite result by complete enumeration is not the same as an empirical, inference to best explanation or generalisation from a consistent pattern, inductive argument. It is like proving a logical conclusion by truth table based examination of cases instead of doing the algebra of logic.

EMH, 11: The material universe is finite and discrete. Mathematics is infinite. Therefore, the material universe is not all that exists. Furthermore, anything that can do math cannot have a material origin. Thus, it is impossible to explain our ability to do math by evolution. This is one of those things that is so obvious materialists just ignore it.

JAD, 12: Go read this there.

BO’H, 13: empirical does not mean “inference to best explanation or generalisation from a consistent pattern, inductive argument”, so you’re shifting the goalposts, and it is nonsense to suggest that complete enumeration means that something is not empirical – “all swans are white” is a statement that can be tested empirically by looking at all swans (if all swans were white!).

KF, 14: . . .  Inferences of inductive character as described are the heart of scientific methods and reasoning. Mathematics, since the days of the ancient Geometers, has been deductive. The number and colour of swans is an indefinite value stretching into the future, the unobserved past and involving unobserved cases in different places — you CANNOT inspect all swans, so inferring whiteness on the pattern was inductive and failed in the 1700’s. A conclusion based on exhaustive inspection of a finite, definite set of cases, is simply not the same as such induction. And note, the issue pivots on reasoning pattern. Mathematics simply does not work in the way Natural Sciences (much less social and psychological/ behavioural ones) do. Indeed, it is the great gap in the naturalistic account, dealing with abstract entities and logical relationships that then by force of how logic affects possibilities and necessities of being, constrain what may be in an instantiated world. The logic of structure and quantity stands athwart the rush of evolutionary materialism and challenges it with cases that cannot be avoided, but whose full significance can be suppressed. For instance, absent a responsible, rational, free, morally governed mind, the integrity of thought and value on truth and doing it right required for Mathematics is fatally undermined. And that brings with it all of that stuff about how can minds be free, and how can minds be morally governed. Where, mere programmed mechanical necessity and/or chance variability do not account for rational, responsible freedom. Cannot account for it. That is where men like Euler speak still, and not just in 0 = 1 + e^i*pi or the like, an expression that shows the deep coherence across huge swathes of Mathematics that in the course of their development did not at all need to come together like that, as far as we know. Which is extremely suggestive on the core nature of the roots of the world.>>

I think this exchange is worth reflecting on. END

Comments
A Mathematical Model reveals complex functionally specified informational complexity. :)
The mathematical model per se is a valuable tool for mathematical and theoretical biologists. Its solution costs very little computationally, so it can be extended to more complex geometries, larger domains, and higher dimensions. By changing the model geometry, its biochemical reaction network, and the model parameters, one can simulate the dynamics of a diverse range of biological phenomena.
A Mathematical Model of Lymphangiogenesis in a Zebrafish Embryo. Wertheim KY, Roose T Bull Math Biol. 79(4):693-737. doi: 10.1007/s11538-017-0248-7
Dionisio
July 17, 2017
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The more we know, more is for us to learn.Dionisio
July 17, 2017
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A Mathematical Model reveals complex functionally specified informational complexity. :)
Even if a species other than VEGFC induces the PCV-derived LECs to differentiate, it probably forms an appropriate concentration profile in one of the scenarios this study proposes. The proposed control mechanisms probably regulate its spatiotemporal dynamics too. More generally, these scenarios and mechanisms are frameworks within which embryonic development and tissue regeneration can be understood.
A Mathematical Model of Lymphangiogenesis in a Zebrafish Embryo. Wertheim KY, Roose T Bull Math Biol. 79(4):693-737. doi: 10.1007/s11538-017-0248-7
Did somebody say "probably"? :)Dionisio
July 17, 2017
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A Mathematical Model reveals complex functionally specified informational complexity. :)
The fact that VEGFC is distributed like a morphogen in multiple scenarios suggests that it is actually one. A different molecular species is probably required to guide the PCV-derived LECs to the horizontal myoseptum in a zebrafish embryo. However, to validate our predictions, we need to comprehend the intracellular responses of an LEC to a concentration gradient of VEGFC.
A Mathematical Model of Lymphangiogenesis in a Zebrafish Embryo. Wertheim KY, Roose T Bull Math Biol. 79(4):693-737. doi: 10.1007/s11538-017-0248-7
Dionisio
July 17, 2017
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A Mathematical Model reveals complex functionally specified informational complexity. :)
The lymphatic system of a vertebrate is important in health and diseases. We propose a novel mathematical model to elucidate the lymphangiogenic processes in zebrafish embryos. The model considers the interstitial flow driving convection, the reactive transport of VEGFC, and the changing dynamics of the extracellular matrix in the embryo. [...] VEGFC behaves very differently in diffusion and convection-dominant scenarios. We predict that VEGFC is a morphogen for the migrating LECs, but it is not a chemotactic factor for them.
A Mathematical Model of Lymphangiogenesis in a Zebrafish Embryo. Wertheim KY, Roose T Bull Math Biol. 79(4):693-737. doi: 10.1007/s11538-017-0248-7
Dionisio
July 17, 2017
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KF, I understand by natural science the research of this universe we are in, with the purpose to gain knowledge about it. Biology is an example. Mathematics is not a Natural Science. It operates mainly in the abstraction realm. However, it has many applications in the natural sciences, like in biology. For example: A Mathematical Model of Lymphangiogenesis in a Zebrafish Embryo. Wertheim KY, Roose T Bull Math Biol. 79(4):693-737. doi: 10.1007/s11538-017-0248-7Dionisio
July 16, 2017
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What is a natural science, why?kairosfocus
July 14, 2017
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KF: Well said, as usual. I must say, however, that I remain happy of my decision not to use the concept of nature in my reasonings. Everything becomes simpler! :)gpuccio
July 14, 2017
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GP, Dionisio and all, please notice my going concern remark above. KFkairosfocus
July 14, 2017
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Dionisio: Well, I understand what he means, even if I don't necessarily agree with his premises. What really surprised me in some interventions here is the idea of starting from the supernatural to define what is natural. I would never have thought of that myself, and I like to be surprised! :) However, I have just published a new OP (not about these matters). You are welcome to read it ! :)gpuccio
July 14, 2017
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gpuccio @65:
"I understand what you say,..."
This is another confirmation that my mind is very slow* to process information. I could not understand what your interlocutor wrote before your comment @65, but you --and probably other people here-- apparently did understand what your interlocutor wrote. To me the comment @64 seems nonsense. I could not honestly say that I understand it. But again, my reading comprehension is rather poor by any standards. (*) see the first sentence @59Dionisio
July 14, 2017
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GP, Maybe AmHD can help insofar as there can be some in common view:
na·ture (n??ch?r) n. 1. a. The material world and its phenomena: scientists analyzing nature. b. The forces and processes that produce and control these phenomena: the balance of nature. 2. The world of living things and the outdoors: spent the day enjoying nature. 3. A primitive state of existence, untouched and uninfluenced by civilization or social constraints: when people lived in a state of nature. 4. The basic character or qualities of humanity: It is only human nature to worry about the future. 5. The fundamental character or disposition of a person; temperament: a man of an irascible nature. See Synonyms at disposition. 6. The set of inherent characteristics or properties that distinguish something: trying to determine the nature of a newly discovered phenomenon. 7. A kind or sort: confidences of a personal nature. 8. a. The processes and functions of the body, as in healing: The doctor decided not to do anything and let nature take its course. b. Heredity: behavior more influenced by nature than nurture. [Middle English, essential properties of a thing, from Old French, from Latin n?t?ra, from n?tus, past participle of n?sc?, to be born; see gen?- in Indo-European roots.]
Other definitions contrast the artificial (as produced by intelligently directed configuration). That goes back to Plato in The Laws Bk X. So, we see right off that the tendency by advocates of evolutionary materialistic scientism and fellow travellers to contrast the supernatural is wrong-headed and a root of much confusion. The natural in this sense speaks of the bio-physical world as a going concern, not reshaped or controlled by art. So, we can look to world as a going concern with dynamics driven by chance and/or mechanical necessity. Redirecting through intelligently directed configuration is an intervention of art and can -- if sufficiently drastic -- destabilise the natural world or at least a relevant part. Thus, we see a three-way analysis, with a natural pair of general types of forces [chance and mechanical necessity] and intelligence acting by art. Further, the going concern view opens up consideration of origins without unduly locking out serious possibilities: the physical cosmos and the world of life possibly were caused in material part through intelligently directed configuration. Where too, said intelligently directed acts can affect particular phenomena even now . . . and often by working with the laws and forces of our world as a going concern, i.e. by technology in the broad sense. Where also that is rooted in the word Plato used which we render "art." (Art seems to be a Germanic term.) Now, we have said nothing here regarding the Supernatural, especially concerning another realm of reality, the spiritual. But obviously that is a serious worldview possibility. arguably, the cosmos itself points that way once we reckon with fine tuning. Similarly, our character as responsibly, rationally free, morally governed creatures is also suggestive. It would seem to me wise not to beg big worldview questions just by how we define nature. Hence, my going concern modifier. Naturalists seem to be arguing that the world we see as going concern is self-explanatory in the causal sense, including origin of intelligent agents. I would like to see actual warrant for such a claim that does not end in self-referential incoherence on accounting for the very mindedness we need to undertake the discussion. I suggest, there are key clues that strongly suggest that going-concern nature is not causally self-explanatory. Going further, there is evidence of miracles, acts that are black swans beyond the usual going concern course of the world. My being alive today is due to one of them. There is responsible reason to hold that we live in a big-C Creation, under God. God would be beyond nature, supernatural by definition. When it comes to acts of art, they seem to be beyond nature going on by its undisturbed self, but whether mindedness is utterly beyond such a world going on by itself would have to be thought through. Similarly, the natural sciences study the world of nature, based on empirical, inductive [modern sense] methods. Mathematics by contrast studies phenomena of structure and quantity, through logic. with emphasis on axionatic systems and a growing body of proved results arrived at deductively through intelligent reflection communicated using symbols and meanings built up as part of the system. This grounds the claim, Mathematics is not a Natural Science. But those locked into Scientism will struggle with that point. KFkairosfocus
July 13, 2017
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Thanks for the discussion, gpuccio.daveS
July 13, 2017
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gpuccio, my first step in developing a common definition would be declaring that "nature" a subset of "reality". It would be like saying the eyes are part of the face but not the face. The second step would be defining nature via negation. What is natural is what is not supernatural. What would be supernatural? They would be the things impossible for man to comprehend. This requires the recognition that there are such things. Examples would be "who made God?"or ad infinitum arguments relating to the cause of The Big Bang. If one believes these questions are answerable by man one would not believe in the supernatural. Those who hold this view, however, express a faith far greater than I can ever dream of having :-)tribune7
July 13, 2017
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daveS: Thank you for your thoughts. :)gpuccio
July 13, 2017
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gpuccio,
Wouldn’t it be better to simply ignore the false problem of nature, and stick to what is real and true?
Probably so. I don't spend a lot of time thinking about the issue (and have about run out of things to say here).
That is fine for you, but a definition of nature that can be used to define what is science cannot rely on what “an orthodox Christian” would know (or, more correctly, believe), I suppose. I don’t know your position, but I don’t do creation science: my science can be certainly influenced by what I believe (that is true for all human beings), but it can never rely on those beliefs.
I agree to some extent. But I suppose if you want to explore what is real, you need to make some metaphysical assumptions, and reason conditionally on those assumptions. I don't think this really contradicts anything you are saying, btw, I'm just elaborating a bit further.
And yet, you could not have done that if you had not represented in your consciousness the form and the purpose of that artifact. It is not different from writing a novel, or a complex software. Either all design is natural or no design is. Either cognition and purpose are natural, or they are not.
Some interesting points, especially the bolded part, IMHO. I'll have to think more about that one.daveS
July 13, 2017
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daveS: "It seems to me you must assume there are no malevolent dieties attempting to confound your attempts to obtain these facts by rigging the vaccine studies. In other words, no dieties which make it impossible to trust induction." Of course. But is is more precise to say that I have absolutely no reasons to assume such arbitrary things. "I would assume that an orthodox Christian would know from the start what beings are in fact supernatural, so I don’t think there is any circularity here. The class of supernatural beings is established first, then their actions are by definition supernatural." That is fine for you, but a definition of nature that can be used to define what is science cannot rely on what "an orthodox Christian" would know (or, more correctly, believe), I suppose. I don't know your position, but I don't do creation science: my science can be certainly influenced by what I believe (that is true for all human beings), but it can never rely on those beliefs. "As far as the generation of biological beings goes, yes, that’s an open question I take it. And perhaps whether human artifacts are natural as well—maybe that’s a choice in definition we have to make. If I created an arrowhead out of obsidian, I would be reluctant to claim that a supernatural event had occurred." And yet, you could not have done that if you had not represented in your consciousness the form and the purpose of that artifact. It is not different from writing a novel, or a complex software. Either all design is natural or no design is. Either cognition and purpose are natural, or they are not. Wouldn't it be better to simply ignore the false problem of nature, and stick to what is real and true?gpuccio
July 13, 2017
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gpuccio,
a) “The things that are real but that do not imply any intervention of God or Satan or other supernatural beings”
I would assume that an orthodox Christian would know from the start what beings are in fact supernatural, so I don't think there is any circularity here. The class of supernatural beings is established first, then their actions are by definition supernatural.
Moreover, do you include among the things that “work something like a clock” the generation of biological beings? I suppose ID is exactly about that. If OOL, or a new protein or species, requires the intervention of a designer, is it natural? New designed things do not come out “like a clock”: they must be designed to exist. Are they natural? Are human artifacts natural?
As far as the generation of biological beings goes, yes, that's an open question I take it. And perhaps whether human artifacts are natural as well---maybe that's a choice in definition we have to make. If I created an arrowhead out of obsidian, I would be reluctant to claim that a supernatural event had occurred.
For example, about your example of vaccines, if I am trying to understand how vaccines perform, I will be interested in anything that really has some role in how vaccines perform. I could well accept any result which is supported by facts: it is not at all likely, but if it could be shown that vaccines have to be blessed to work, and if observed facts really did support that conclusion, I would not say that the conclusion is not scientific because blessing is not a part of nature. I would really try to understand what is at work there.
The point I had in mind related more to how you obtain these facts. It seems to me you must assume there are no malevolent dieties attempting to confound your efforts to obtain these facts by rigging the vaccine studies. In other words, no dieties which make it impossible to trust induction.daveS
July 13, 2017
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daveS: I understand what you say, but frankly defining nature as: a) "The things that are real but that do not imply any intervention of God or Satan or other supernatural beings" sounds a little bit self-referential, and not completely scientific :) Defining natural starting from the supernatural could not be a very good idea, after all. Moreover, do you include among the things that "work something like a clock" the generation of biological beings? I suppose ID is exactly about that. If OOL, or a new protein or species, requires the intervention of a designer, is it natural? New designed things do not come out "like a clock": they must be designed to exist. Are they natural? Are human artifacts natural? Believe me, ID (and science, for that) can live very well without any specific need to refer to "nature". reality is more than enough. For example, about your example of vaccines, if I am trying to understand how vaccines perform, I will be interested in anything that really has some role in how vaccines perform. I could well accept any result which is supported by facts: it is not at all likely, but if it could be shown that vaccines have to be blessed to work, and if observed facts really did support that conclusion, I would not say that the conclusion is not scientific because blessing is not a part of nature. I would really try to understand what is at work there. Science can have no preclusions, but it must stick to facts. There is no evidence that vaccines need to be blessed, and a lot of evidence that they work in a very different way. We always look at the best inference. But "naturalism" is not a good idea to understand what best inference can be considered science, and what should be dogmatically excluded.gpuccio
July 13, 2017
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gpuccio, Maybe it would help us to get a clearer definition of "supernatural" (and thus of "natural") if we focused on a particular worldview. For example, in orthodox Christianity, we could say a supernatural event is an event that involves some overt intervention of God or Satan (or perhaps angels, demons, &c.). I'm assuming here that there may be parts of reality where, although God is ultimately responsible for their existence, they work something like a clock, and where God or some other power does not have to continuously intervene.daveS
July 13, 2017
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gpuccio @ 62: Well done.Truth Will Set You Free
July 13, 2017
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daveS and tribune7: Interesting interventions, but again they make me think that "nature" can be defined in many bizarre and different ways, none of which really shared by others. Would you try your own definition? I will try a few possible ones: "What really exists" (but then it is the same with "reality", which, as tribune7 says, is a much better term. I have no objections to reality, and I would definitely embrace any form of realism, methodological or not. :) Now, if nature is not reality, that means that in reality there are things that are not in nature. But then, what is the difference? We have already found some interesting cases, and I will mention some: a) Our personal consciousnes b) The number 1 c) Dark energy (if, as I believe, it exists) Now, what kind of definition of the "natural" subset will give us clear judgments for that kind of things? Most would say that dark energy, whatever it is, is natural, but why? We have no idea of what it could be. Number 1 seems to be at the center of hot discussions here. What about our personal consciousness or, for those who believe in it, out personal soul? Let's try some definitions: a) Nature is what is real and material. But what is matter? Not so easy... Is energy in nature? Are wave functions in nature? Is consciousness in nature? b) What can be explained by the general views of the universe we have today. That is probably what most materialists mean, and here is the trick. I just ask: why? Why should what we know today define nature? Was that true a couple of centuries ago, too? Say before quantum mechanics and relativity? Would a methodological naturalist, at that time, have stated that science could only accept as explanations those that were set in a fixed time and space? Would a mathematician, before Godel, have stated that mathematical nature (if we accept such a concept) could only accept systems that were included in the universal formalization of mathematics (still to be found at that time, never to be found after Godel)? You seem to think that I hate the word "nature" because I want to include supernatural entities in my reasonings. But that's not the case: I hate the word "nature" because I don't know what it means. To be sincere, I have the same bad feelings for the word "supernatural", and for the same reasons.gpuccio
July 13, 2017
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Error @59: the text: "...so it does not interferes with..." should read: "...so it does not interfere with..." Sorry for the mistake. My fault. BTW, is the following article related to the ongoing discussion? Mathematical Biology is Good for Mathematics Michael C. Reed DOI: 10.1090/noti1288 http://www.ams.org/notices/201510/rnoti-p1172.pdfDionisio
July 13, 2017
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gpucci, I think "nature" is a very useful word in context. The problem is some have made it synonymous with "reality" which is not what it was meant to mean. There is part of reality where there are consistencies. To call this part "nature" is good. When studying these consistencies one will obviously limit the study to those consistencies. The problem comes when we insist that there is nothing but consistencies, and we adapt the methods of explaining consistencies to all aspects of our existence such as love, purpose and justice. While studying nature can be useful in some aspects of these aspects, to make it a definitive arbiter is the path to complete misery, not to mention that it is logically inconsistent.tribune7
July 13, 2017
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I've enjoyed this interesting discussion thread, though the exchange of comments seems faster than my mind can keep up with. As I said at the beginning of this thread, I like mathematics since I was a teenager and have memories of my learning of calculus in high school. Also, one of my best friends from that time is addicted to high mathematics, having a PhD from the University Lomonosov in Moscow and teaching at another university. In high school he played chess blindfolded and still won easily. Obviously, his opponents were no match for him. As far as I understand it, the core of mathematics operates mainly in the abstraction realm, hence it proves itself. However, it has many applications in most fields of science and beyond. Its applications in Biology keep growing unstoppably. Actually, these days all the fields of serious science are involved in helping the biology researchers to understand the functioning of the wonderfully designed biological systems. Not sure if this is "in topic", but here's an article that may illustrate the wide variety of applications of mathematics in biology-related research:
Mathematical Modeling of Prion Disease By Suzanne S. Sindi DOI: 10.5772/66917 Online article: https://www.intechopen.com/books/prion-an-overview/mathematical-modeling-of-prion-disease PDF: https://cdn.intechopen.com/pdfs-wm/53804.pdf Published as Chapter 10 in the book:
Prion - An Overview Edited by Yusuf Tutar, ISBN 978-953-51-3002-4, Print ISBN 978-953-51-3001-7, 236 pages, Publisher: InTech, Chapters published March 08, 2017 under CC BY 3.0 license DOI: 10.5772/63289 https://www.intechopen.com/books/prion-an-overview
[emphasis added] If this is off-topic, please remove this comment so it does not interferes with the flow of the ongoing discussion. Thanks.Dionisio
July 13, 2017
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gpuccio, I would agree to the statement that "nature" is a vague term, certainly, and hence so is "methodological naturalism". I wouldn't conclude that the concept of MN is somehow a "trick", or even a meaningless or useless concept, however. That said, I'm not very familiar at all with the debates over MN, so what follows is just me thinking out loud. How do you approach MN as a physician? Suppose you were in charge of trials of a vaccine for some serious disease. Surely you would expect the results of your trials to reasonably represent the results of future use of the vaccine, as long as the trials were well designed. You would not expect some malevolent deity to interfere, say by making the vaccine look safe and effective in trials, but then refrain from interference once the vaccine was adopted, in which case the vaccine is 100% fatal. Even assuming you agree with the above, obviously this doesn't mean you deny the existence of supernatural deities. I guess one could still maintain that literally every event that occurs (including events in the vaccine trial) are caused by a deity. However, I'm guessing that at least some theistic physicians appeal to "natural" causes, at least as a meaningful shorthand for _something_, even in cases where there is supernatural involvement.daveS
July 13, 2017
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KF at #55: Very useful, thank you.gpuccio
July 13, 2017
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tribune7, daveS: Your interesting exchange just reinforces my old conviction: the word "nature" has no well defined meaning at all. That's why so called "methodological naturalism" is just a trick for materialist propaganda.gpuccio
July 13, 2017
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F/N: Wiki on mathematical Induction:
Mathematical induction is a mathematical proof technique used to prove a given statement about any well-ordered set. Most commonly, it is used to establish statements for the set of all natural numbers. Mathematical induction is a form of direct proof, usually done in two steps. When trying to prove a given statement for a set of natural numbers, the first step, known as the base case, is to prove the given statement for the first natural number. The second step, known as the inductive step, is to prove that, if the statement is assumed to be true for any one natural number, then it must be true for the next natural number as well. Having proved these two steps, the rule of inference establishes the statement to be true for all natural numbers. In common terminology, using the stated approach is referred to as using the Principle of mathematical induction. Metaphors can be informally used to understand the concept of Mathematical induction, such as the dominoes falling in a line, or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the induction). —?From the Margins of the book, Concrete Mathematics, page 3. The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction, in some form, is the foundation of all correctness proofs for computer programs.[4] Although its name may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning (also see Problem of induction). Mathematical induction is an inference rule used in proofs. In mathematics, proofs including those using mathematical induction are examples of deductive reasoning, and inductive reasoning is excluded from proofs.[5]
Does that help to clarify? KFkairosfocus
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RVB8, It may help you to know that GP is a physician. As he just pointed out, I see. KF PS: Creationism and its alleged lack of interest in how or principles of Creation are irrelevant to ID. In fact, it might help you to correct your thoughts to understand that major early modern scientists saw themselves as thinking God's thoughts of creation and providence after him, as part of the stewardship of creation. You seem consitutionally incapable of acknowledging that design thinkers and theorists are interested in and involved in research, including on design. Further, that there is now a significant body of professional literature from such theorists, and a wider body that is relevant and even often supportive though not under that school of thought. Accountability before evident facts is a key step in intellectual growth.kairosfocus
July 13, 2017
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