At Mind Matters News: Why would a purely physical universe need imaginary numbers?

_{Denyse O'Leary

February 17, 2022

Intelligent Design, Mathematics}

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Our computers and the entire modern world depend on them, says science writer Michael Brooks in an excerpt from his new book:

In an excerpt from his new book, The Art of More: How Mathematics Created Civilization, science writer Michael Brooks offers the intriguing idea that the modern world arose from imaginary numbers:

But what does his claim that the numbers are “not some deep mystery about the universe” leave us? Recent studies have shown that imaginary numbers — which we can’t really represent by objects, the way we can represent natural numbers by objects — are needed to
describe reality. Quantum mechanics pioneers did not like them and worked out ways around them:

In fact, even the founders of quantum mechanics themselves thought that the implications of having complex numbers in their equations was disquieting. In a letter to his friend Hendrik Lorentz, physicist Erwin Schrödinger — the first person to introduce complex numbers into quantum theory, with his quantum wave function (ψ) — wrote, “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. Ψ is surely fundamentally a real function.”
Ben Turner, “Imaginary numbers could be needed to describe reality, new studies find” at LiveScience (December 10, 2021)

But recent studies in science journals Nature and Physical Review Letters have shown, via a simple experiment, that the mathematics of our universe requires imaginary numbers.
News, “Why would a purely physical universe need imaginary numbers?” at Mind Matters News (February 16, 2022)

Takehome: The most reasonable explanation is that the universe, while physical, is also an idea, one that cannot be reduced to its physical features alone.

You may also wish to read:

Why the unknowable number exists but is uncomputable. Sensing that a computer program is “elegant” requires discernment. Proving mathematically that it is elegant is, Chaitin shows, impossible. Gregory Chaitin walks readers through his proof of unknowability, which is based on the Law of Non-contradiction.

Most real numbers are not real, or not in the way you think. Most real numbers contain an encoding of all of the books in the US Library of Congress. The infinite only exists as an idea in our minds. Therefore, curiously, most real numbers are not real. (Robert J. Marks)

and

Can we add new numbers to mathematics? We can work with hyperreal numbers using conventional methods. Surprisingly, yes. It began when the guy who discovered irrational numbers was—we are told—tossed into the sea. (Jonathan Bartlett)

Comments

KF, when I wrote, “we can’t know about the root of reality seems much more likely, based on evidence, than thinking we can ... “ You wrote, “See the self referential problem there, even with the relative likelihood claim?” No, I do not. I have many times discussed the provisional nature of my claim, based on empirical evidence about the nature of human belief as shown through studies of comparative religion, philosophy, psychology, etc. You dismiss those and seem to have no understanding at all of the distinction between my statement and a “universal negative claim” that you later say I am making. You later say, “If your effective quarrel is with these [the laws of logical reasoning], your problem is with reason not my alleged assumptions.” Once again, you PAY ABSOLUTELY NO ATTENTION TO WHAT OTHER PEOPLE ACTUALLY SAY!!! Pardon my shouting, but I’ve said multiple times that I accept that the laws of logic are an essential part of our rational cognitive abilities. It would be really nice if you would acknowledge that so you don’t keep bringing up the same irrelevant issue of that being in question. Then you write, “As to science is all we got, that seems to be a summary from SA perhaps paraphrasing, it points to scientism, ... “ Again, you ignore what others are saying. Neither I, WJM, or Ram are advocating for anything approaching scientism. Then you write, “As for rooting of the world, it is notoriously temporal-causal and successive, one year follows another for instance. It is therefore appropriate to consider where it came from. An initial candidate is, infinite past. “ As I and WJM have pointed out a number of times, space, time, and linear causality are features of our world. Thinking that they as we know them apply to whatever exists “outside” or “before” our world (both words themselves being inappropriate but we have no other words to use) is not justified. The general problem with your “self-referential incoherence” argument is that you always judge statements by others by your own philosophical framework, so anything anyone says that comes from a different framework and contradicts your framework is labeled “incoherent”, Yes, it’s “incoherent” to you because you have such a rigid, narrow view in which only your perspective is true. It’s one thing to disagree with people. It’s another thing to be unable to really understand what other people are and are not saying and to be so sure of yourself that anyone who disagrees with you, working from a different perspective, is guilty of “self-referential incoherence.” That’s why I called your frequent claim of self-referential incoherence a “ploy”. It’s a tactic to avoid recognizing and responding to a different view than your own.Viola Lee_{February 23, 2022
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Ram, kindly note my two comments just above. KFkairosfocus_{February 23, 2022
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SA, yes, but these days such is marginalised by the credentialed and chattering class. It turns what should be generally recognised into needless controversy. This then clouds onward thinking. Aquinas' little errors in the beginning at watershed lines in our thinking ending up in oceans of error a continent away from otherwise readily knowable truth. That is why we need to be so careful to recognise first principles of reason antecedent to proof as such already bake them in. KF PS, Epictetus,
DISCOURSES CHAPTER XXV How is logic necessary? When someone in [Epictetus'] audience said, Convince me that logic is necessary, he answered: Do you wish me to demonstrate this to you?—Yes.—Well, then, must I use a demonstrative argument?—And when the questioner had agreed to that, Epictetus asked him. How, then, will you know if I impose upon you?—As the man had no answer to give, Epictetus said: Do you see how you yourself admit that all this instruction is necessary, if, without it, you cannot so much as know whether it is necessary or not? [Notice, inescapable, thus self evidently true and antecedent to the inferential reasoning that provides deductive proofs and frameworks, including axiomatic systems and propositional calculus etc. We here see the first principles of right reason in action. Cf J. C. Wright]
kairosfocus_{February 23, 2022
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VL, Let's note one of your statements at outset, so we can pin down the matter, this being perhaps your most radical assertion:
we can’t know about the root of reality seems much more likely, based on evidence, than thinking we can
See the self referential problem there, even with the relative likelihood claim? A self referentially incoherent claim is instantly of zero likelihood as it cannot be true. What cannot be true is only knowable as necessarily false. Now, I am not making a priori metaphysical commitments of any consequences; despite your assertions to the contrary. Where, your repeated assertion that I have done so without specifying what I have allegedly assumed that I should not becomes questionable. It raises the issue of an implicit ad hom circumstantial, i.e. you know I happen to be a Christian -- an end point of reasoning and experience, not an inherited assumption -- and may be implying inevitable defective reasoning for anyone with such views. But in fact the reasoning I am using is general. Did I merely assume first principles of right reason or impose them on mere priestcraft? Not at all. For, first principles of right reason are self-evident, pervasive and inescapable, branch on which we all sit stuff. Distinct identity is necessary to think and communicate, as was noted from a C1 Rhetoric 101 example; admittedly I used Paul but merely as an apt cite on a general fact. As in, what are these distinct glyphs strung together to make meaningful text about? To evade this, you are stuck with silence and non thought, then the difference and identity of food, drink, clothing and shelter stare you in the face. Those who try to live apart from this instantly fail. As you know or should acknowledge, non contradiction and excluded middle are close corollaries. If your effective quarrel is with these, your problem is with reason not my alleged assumptions. As to science is all we got, that seems to be a summary from SA perhaps paraphrasing, it points to scientism, that science so monopolises or dominates knowledge that anything otherwise rooted doesn't count. As I have noted, this is an epistemologicsl, i.e. philosophical claim and refutes itself. In any case, science depends on Math, which does not work on empirical but logical methods driven by at basic level first facts, at another, axioms derived from framing such facts then setting up logic model worlds. Going on, reductio is an application of non contradiction, thence principle of explosion. H being asserted, consequences are derived, and once a contradiction or denial of known fact results, fail, so the denial ~H is true. In your case, you keep asserting un-know-ability of the root of reality. As I pointed out yesterday at 281, many times in the course of an argument. But that is itself a knowledge claim and it about said root, it fails. As for rooting of the world, it is notoriously temporal-causal and successive, one year follows another for instance. It is therefore appropriate to consider where it came from. An initial candidate is, infinite past. And even this is knowledge about the root, that such is a candidate. It turns out, traversal of an explicit or implicit transfinite span in finite stage steps such as years is an infeasible supertask. So, my point is, had you instead said unknown to you or controversial among thinkers, yes obvious. Philosophy being the study of hard questions (towards a coherent understanding of reality), that is the case inevitably. So, comparative difficulties across factual adequacy, coherence and explanatory power. But that is not what you have said. You have been asserting un-know-ability of the root of reality, a universal negative claim that implicitly is itself a knowledge claim. So, it refutes itself. To see that we don't need anthropology etc, we just need to recognise a knowledge claim and when it becomes self referential and incoherent. KFkairosfocus_{February 23, 2022
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VL: Who here is saying “Science is all we got?” Certainly not me, WJM, or Ram. Indeed. --Ramram_{February 22, 2022
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Who here is saying "Science is all we got?" Certainly not me, WJM, or Ram. Also, FWIW, I accept that the laws of logic are essential for human understanding. I don't accept, however, many of the assumptions and conclusions KF reaches: much of what he considers as purely logical is infused with his own metaphysical assumptions.Viola Lee_{February 22, 2022
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KF
A better approach is to start with highly knowable self evident first principles of reason then define knowledge in that light and apply logic to reality, being and its roots. Especially to possible vs impossible being, contingent and necessary being further informed by possible world speak.
That's a very nice summary. You've done several articles that move through those steps. The guy here who says "science is all we've got" is badly confused by scientism. You can't do anything without the first principles and then analysis of reality in light of that.Silver Asiatic_{February 22, 2022
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VL, Pardon, but we don't get to have our cake and eat it: >>You write, “One thing that is clear from the outset is to assert that the root of reality is unknowable is to assert a knowledge claim about said root>> 1: Manifestly the case, despite your claims to the contrary. 2: Had you instead said, unknown to me, or controversial among scholars etc -- everything in Philosophy is controversial as hard questions have no easy answers and so we work by comparative difficulties -- that would have been different. But, that is NOT what you have claimed. 3: That's why I called you on manifest self referential incoherence thus principle of explosion and reductio. >>and it is thus self referentially incoherent and falsifies itself”>> 4: Yup. >>This is your favorite ploy.>> 5: In effect you are trying to use loaded language to dismiss a logical issue. Fail. >> What you fail to acknowledge is my repeated disclaimer that I am not asserting something I know to be true:>> 6: So, do not assert un-know-ability, instead say unknown to me or subject to controversies under comparative difficulties. As was already noted. >> I am making a claim that taking all I know about humans beings, including about religion, psychology, philosophy, anthropology, etc,>> 7: Do you see the little knowledge claim there? That is immediately an illustration of claiming to know un-know-ability. >> all of human beings metaphysics are inventions,>> 8: Metaphysics is a study, of a substance involving critical examination of worldviews, towards synthesis, and as such is a discipline with 2400+ years of history going back to Thales on the dock at Miletus pausing between futures investments to ponder the one and the many. 9: The substance is being, reality, its contents and their roots. As I summarised above, R --> W etc. >> so I don’t think any of them is “true”.>> 10: Here, denial of our ability to overcome error and through right reason find warranted truth, of course itself being precisely a worldview by the backdoor, hyperskepticism influenced radical relativism, maybe the relativism of one, subjectivism. 11: A worldview claiming to be true and so refuting itself by self referential incoherence. >> I have other reasons for thinking that we really can’t know about the root of reality, but they are all provisional,>> 12: See the repetition of the un-know-ability thesis regarding roots of reality, claiming to be knowledge regarding said topic? 13: Doubling down on self referential absurdities is question begging but can be highly persuasive. >>but it seems that saying we can’t know about the root of reality seems much more likely, based on evidence, than thinking we can.>> 14: More of the same incoherent circularity. 15: A better approach is to start with highly knowable self evident first principles of reason then define knowledge in that light and apply logic to reality, being and its roots. Especially to possible vs impossible being, contingent and necessary being further informed by possible world speak. 16: Much, as has been done above and elsewhere. KFkairosfocus_{February 22, 2022
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Ram, toxic, red herring based distractor; especially as you know that UD is not a theology blog with a resident panel of experts and one of the slanders out there is that design theory is creationism in a cheap tuxedo. See our Weak Argument Correctives under the resources tab. Thus, as you have been advised on the point repeatedly, insistence on such behaviour is, regrettably, a sign of want of good faith. The intent, manifestly, is to switch focus from something YOU are uncomfortable with and project. Further to which, you refuse to recognise that what I have discussed above and in other threads has little or nothing to do with Bible Study or Biblical Theology. It has everything to do, instead with [a] empirical investigation of reliable signs of design in the world of life and the cosmos (tracing to considerations of inductive logic and epistemology), and [b] investigation of logic of being tied to root of reality, i.e. that worldview integrative exercise known as Metaphysics and particularly Ontology. As can be seen from 116 above, the issue here has been the claim that root of reality issues are "unknow-ABLE," where that claim is self referential, implies it is objective and known, refuting itself. Under normal circumstances, that would suffice to settle the matter, as incoherence due to self referential self contradiction is reduction to absurdity, strong form. But these are not normal times. Perhaps, it has not dawned on you that I have not set out on exposition and systematisation of scriptures but on examination of empirical evidence and on worldviews analysis as appropriate. Here, spending a fair bit of effort on foundations of Math, responsive to Wigner's wonderment over the power of Math in science, tracing to the concept of logic of necessary being -- ontology -- and examples of knowledge of roots of reality. Where BTW, philosophy in this context has a senior claim to science. In particular, scientism is self referentially incoherent right from the term itself. That is, the notion that science monopolises or so dominates serious knowledge claims that its voice of the moment is the decisive queen move, is a claim in EPISTEMOLOGY, a core branch of . . . you guessed it . . . Philosophy. Hence, Scientism is absurdly incoherent, but exploits institutional domination to impose agendas that would not otherwise prevail. As we have seen at first hand with marginalisation and denigration of a growing number of effective treatments since March 2020. Sciences and technologies are useful in their place, as is fire. But out of place, they help induce marches of ruinous folly. As for your reconstruction of origins of modern science, it reeks of revisionism and promotion of agendas seeking to ride on the tails of the lab coats of scientism. Fail. KFkairosfocus_{February 22, 2022
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Hey, BA77 and KF and fellow travellers... Do you believe that a literal serpent talked to a literal Adam and Eve in a literal garden about 6000 (give or take) years ago? Yes or no? Thanks (I wonder, as I ask the question, how embarrassed they must feel about the question, and how threatened they must feel about the question. But why feel embarrassed or threatened if reality is on your side?) Now... if you don't literally believe that happened, then, well, by golly, there's something to talk about. Right? I can hear it now.... that's a diversion. Yeah, well, anything KF doesn't like facing up to is a diversion. Funny how talking about their beloved scriptures is always some kind of diversion. Is that what Jesus would say? Yeah, yeah, this isn't a Bible study. But... it really is, at the end of the day. Take away their religion/scriptural-interpretation, and they are left with what everyone else is left with: science. And believe me, they will whine, moan and complain, but they love science and rely on it over all as well as the rest of us. And they know this. Who do they think they're kidding? And modern science was delivered by occultists, kabbalists, and homosexuals. Living in the world of modern dentistry is a wonderful thing! Just try living without it when you're in a pinch. You'll trade your soul to satan for some sweet relief. :D --Ramram_{February 21, 2022
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VL and WJM... Nice to have some fresh air around here. Kintergardeners often hate giving up their li'l philosophical blankies. All in good time. Peace. --Ramram_{February 21, 2022
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BA77: The inductive methodology of Francis Bacon, i.e. the scientific method itself, literally has the presupposition that science can answer the question of the ultimate nature and/or truth of reality built into it. It was in spite of Christianity not because of it. Dum dee dum dum. --Ramram_{February 21, 2022
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KF: Ram, pardon, your prejudice is showing. You were answered on that at 123 above, but of course ignored. KF Blah blah blah. Not persuasive. Are you saying your god is a bronze-age god? Is that what you're defending? --Ramram_{February 21, 2022
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VL @ 274 Hold on to those "self-referential incoherencies." They can be redeemed at the Discovery Institute's Discovery Store (for real) for things like coffee mugs, stylish tote bags, ball caps, and tons more cool stuff...... :-)chuckdarwin_{February 21, 2022
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KF writes, " I have worked through the logic." And not everyone agrees with the assumptions you make and the conclusions you reach. I certainly don't. You write, "One thing that is clear from the outset is to assert that the root of reality is unknowable is to assert a knowledge claim about said root and it is thus self referentially incoherent and falsifies itself" This is your favorite ploy. What you fail to acknowledge is my repeated disclaimer that I am not asserting something I know to be true: I am making a claim that taking all I know about humans beings, including about religion, psychology, philosophy, anthropology, etc, all of human beings metaphysics are inventions, so I don't think any of them is "true". I have other reasons for thinking that we really can't know about the root of reality, but they are all provisional, but it seems that saying we can't know about the root of reality seems much more likely, based on evidence, than thinking we can. Your "self-referential incoherency" argument just doesn't wash. That's what I think.Viola Lee_{February 21, 2022
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WJM, the creation of our space-time manifold is not an incoherent concept. It may seem strange but even the concept of a sub cosmos with fluctuating universes is like that. KFkairosfocus_{February 21, 2022
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Sev, I extend the quote:
1 Cor 13: 9 For we know in part and we prophesy in part, 10 but when the perfect comes, the partial will pass away. 11 When I was a child, I spoke like a child, I thought like a child, I reasoned like a child. When I became a man, I gave up childish ways. 12 For now we see in a mirror dimly, but then face to face. Now I know in part; then I shall know fully, even as I have been fully known. [Paul of Tarsus, c 55 AD]
And yes, that is a foundational Christian text. KFkairosfocus_{February 21, 2022
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VL, there is a big gap between merely thinking one knows and working through the logic of being. I have worked through the logic. Others are showing you some of it. One thing that is clear from the outset is to assert that the root of reality is unknowable is to assert a knowledge claim about said root and it is thus self referentially incoherent and falsifies itself. From there we can work out positive things we know. KFkairosfocus_{February 21, 2022
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WJM
Also, as a reminder, the idea of “a creator” of space-time is logically self-contradictory.
Once we move past scientism and materialism and begin talking about the nature of God, we accept that there are paradoxes. Your worldview has them, certainly. But we just try to present what is most reasonable - even though we bump into various limits and obscurities. That reality has a source or origin aligns with the concept of a creator. That our lives had a beginning is intuitive. That the universe had a beginning is a scientific proposal. So, where there's a beginning, there's an act - call it creation, call it actualization. We could say "we don't know what to call it" but I don't see that as different from saying "it's analogous to the human design process - not the same, but like it enough that we use the word 'create'." Communication on this topic would be impossible if we had to have exact, univocal terms. That's an option. "This is a topic I cannot talk about because we don't have human terms for it". Ok, I could go with that. The person just exempts himself from any discussion since it's all "unknowable and beyond comprehension". But I think we try to discover things and it's good to discuss what we can find. To do that, we use terms like "doing" or "creation". We can just say upfront: "it's not an exact match with what God actually does" - then just go forward from there. As long as a person doesn't think that God is trapped by time or limited by material resources in what He can create, it doesn't seem to be a problem. True - some people could get confused by that and picture God as a material being, but that's can be easily corrected.Silver Asiatic_{February 21, 2022
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WJM
One cannot give what one does not have – thus, the origin of all that, possesses all that.
Whatever is all being cannot be "a composition".Silver Asiatic_{February 21, 2022
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SA said:
By logic, the creator cannot be composed of what has been created.
Didn't you just say:One cannot give what one does not have – thus, the origin of all that, possesses all that. Color me confused. Also, as a reminder, the idea of "a creator" of space-time is logically self-contradictory. So is the idea of making a decision "outside of space-time." Any personal experiences expressed as a feeling of being beyond or outside of time and space sure didn't happen outside of their linear space-time experience or else they wouldn't know when it occurred or that it happened to them. Nor would they have been able to experience any sequences of qualia. All of that happens within a framework of at least personal space-time. The act of creation or making a decision any any normative sense requires at least a personal sense of linear time. Yes, there is logic that extends from the assumption of an original creator making choices and creating things, but it's that assumption that I question because I don't see how that is a rational perspective.William J Murray_{February 21, 2022
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WJM
I can’t make sense of this
It's not my personal idea, it's just an adaptation of the classic philosophical argument on the degrees of perfection.Silver Asiatic_{February 21, 2022
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WJM
We can’t experience that. It’s a logical inference. It may be a logical necessity. But, it’s not based on any direct observation or knowledge of that source.
Yes, agreed. We take what we know and what we can derive by logic and arrive at knowledge. But also, we can reach a high degree of certainty about things even without direct experiential knowledge. You gave an analogy on cooking to beef wellington. I see it more like like my kitchen as an analogy for all the kitchens in the neighborhood. I see I have a stove, so with no direct experience of any of those kitchens, I conclude that most of them probably have a stove. But we see certain values within our universe and we infer that they have an origin in a power that can create those values. By logic, the creator cannot be composed of what has been created. It's not a strict deduction but it's "friendly to reason" to conclude that the creator has knowledge of what was created and the act of creation is something the human mind could recognize as such, even though its not done by material means or in sequential time. As I said before, humans experience timeless moments and things that transcend the linear, sequential movements. Even acts of great creation happen in a timeless, spaceless sort of way. Mozart created an entire symphony in his mind in a matter of minutes. This is the most incredible example I've ever seen of the God-given power in a human being to understand far beyond what we would think possible. I can't resist yet another kick at the dead horse - this video renders Darwinism a stone-age collection of myths. The complete ignorance of science on these matters is astounding: Brain Man: The Boy With The Incredible Brain (Superhuman Documentary) | Real Stories https://www.youtube.com/watch?v=PPySn3slfXI 11 million views so you might have already seen itSilver Asiatic_{February 21, 2022
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SA said:
One cannot confer non-being or create the non-existent.
We've been over this. All possibilities are always present in potentia.
Deficiencies are a lack of perfection – they’re not a positive creation of being. Fullness of existence is absolute perfection because it lacks nothing of the good, true and beautiful. A lack in any of those things is merely a potential for perfection, not a positive creation of non-existence.
I can't make sense of this other than to chalk it up to your personal ideology.William J Murray_{February 21, 2022
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^^^^^ Cannot be 'too careful' what one reads eh VL?
“In reading Chesterton, as in reading MacDonald, I did not know what I was letting myself in for. A young man who wishes to remain a sound Atheist cannot be too careful of his reading. There are traps everywhere — "Bibles laid open, millions of surprises," as Herbert says, "fine nets and stratagems." God is, if I may say it, very unscrupulous.” - C.S. Lewis, Surprised by Joy: The Shape of My Early Life
bornagain77_{February 21, 2022
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re 256: BA, I invite you to not read my posts. I don't read yours.Viola Lee_{February 21, 2022
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KF @ 246 Some of us, perhaps. Some of us see reality only too well...chuckdarwin_{February 21, 2022
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SA, you write,"VL: I’m not offering this Taoist perspective as true, SA: Ok, so you think it is not true. It is therefore false." As I've said, we have an unbridgeable gulf between us. Our metaphysics are inventions. They are like literature, which is not intended to be true but is meant to play some significant role in our understanding. And WJM did a much better job at 257 of replying to your post at 251. I'm willing to call our discussion a dead-end, also.Viola Lee_{February 21, 2022
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WJM One cannot confer non-being or create the non-existent. Deficiencies are a lack of perfection - they're not a positive creation of being. Fullness of existence is absolute perfection because it lacks nothing of the good, true and beautiful. A lack in any of those things is merely a potential for perfection, not a positive creation of non-existence.Silver Asiatic_{February 21, 2022
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Also, referring to the ground of being is not predicated on knowing anything about the ground of being; it is a logical inference derived from our own existential position and capacity to reason. We can't even tell that it is emanating us. We can't see that happening. We can't experience that. It's a logical inference. It may be a logical necessity. But, it's not based on any direct observation or knowledge of that source.William J Murray_{February 21, 2022
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