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Question of the Month

What Are The Limits of Knowledge?

Each answer below receives a book. Apologies to the entrants not included.

To answer this question, first ‘knowledge’ must be defined. It is difficult to determine exactly what constitutes knowledge, so even this presents a limit to knowledge.

We might identify various forms of knowledge, such as declarative knowledge (knowing that), procedural knowledge (knowing how), personal knowledge (of experience, emotions) and shared knowledge (true ideas widely accepted in communities and cultures). Sir Karl Popper developed a theory of three worlds, in which World 2 is all subjective (personal) knowledge, while World 3 is all knowledge existing independent of individual minds, such as stories, theories, mathematical constructs, scientific concepts, cultural beliefs, and intellectual creations.

However, categorising knowledge into types does not tell us what it means to know something. One widespread philosophical definition of knowledge is Socrates’ idea of ‘Justified True Belief’ (JTB). This holds that to know some proposition p, the knower must believe that p is true, p must be true, and there must be good justification (good reasons, evidence, or argument) for believing p to be true. However, the problem with JTB is that justification is difficult to establish with certainty. This is shown in Gettier cases; Edmund Gettier demonstrated that some justifications for true beliefs may be a matter of luck, so that an apparently good justification for a true belief may be actually bad; for example, a person declares that it is 2pm (belief), and it is in fact 2pm (true), but his belief is based on seeing a clock that happens to have stopped ticking at 2am (justification). The core of the problem is knowing for certain that our justifications are good enough. This presents a limitation to our concept of knowledge itself.

Popper famously defined the limits of knowledge obtained through empirical (scientific) methods: his ‘falsification’ hypothesis states that genuine scientific theories can be falsified, that is, shown to be false. No matter how many white swans we observe, the claim ‘all swans are white’ requires only one black swan to falsify the claim. So, the only certain statement here is that ‘It’s possible not all swans are white’. Popper argued that our negative findings stimulate further enquiry, leading to further knowledge. Thus, according to Popper, knowledge is always incomplete. In which case, the quest for knowledge is surely limitless. As Einstein is quoted as saying, “The more I learn, the more I realize how much I don’t know.”

Martin Barge, London

Ne plus ultra is Latin for ‘nothing further beyond’. It is said to have been inscribed on the Pillars of Hercules at the western end of the Mediterranean, then the boundary of the known world, as a warning to ships to sail no further. In The Critique of Pure Reason (1781), Immanuel Kant invokes the Pillars of Hercules for the boundary of knowledge and warns that the ‘no further’ must be placarded on the Pillars of Hercules that “nature has erected, so that the voyage of our reason may proceed only as far as the continuous coastline of experience reaches.”

Before Kant, the assumption was that knowledge must conform to objects of sense experience. Instead, Kant argues that the objects of sense experience must conform to the cognitive faculties of the mind of the experiencer. The world as given to our experiences is dependent on the structure and activity of our minds. For instance, the fact that we cannot perceive anything that exists outside of time or which has no extension in space indicates that time and space are fundamental forms of perception that must already exist as innate structures for the human mind. The mind must also have preconscious categories of understanding and synthesis that apprehend and reproduce the raw data of sensory experience and recognize their features according to a conceptual framework. And since everything we perceive is filtered through the forms of space, time, and other categories of understanding, we can never really ‘know’ the actual (noumenal) world – the things in themselves – but only experience how they appear to us through these categories. So these categories are the necessary conditions of all experience.

Since human beings can know reality only by using the human conceptual framework and cannot step outside the limitations of our experience, our knowledge is inevitably limited. Therefore, as Kant said, we must not continue the voyage of our reason beyond the shores of experience, and cease all attempts to gain knowledge of such things as the intrinsic nature of the universe, the existence of God, self, free will and immortality. Going further will be perfectly hopeless, as we will be lost at sea.

Nella Leontieva, Randwick, New South Wales

The first recipient of a PhD in Art Education in the UK was also its first professor in the field. The award represented a new area of knowledge, not covered by its separate components. But this sort of recombination of fields is far from unprecedented. Indeed, knowledge continues to grow and expand within unitary areas, within combined ones, and across both. Melvin Bragg, presenter of BBC Radio 4’s In Our Time, recently referred to this “greatest age of expanding knowledge, a tumult of erupting knowledge.” Its expansion supports the contention that there is no foreseeable limit, at least to what we can know, unless we reach a point where all that we can know, is known. Beyond this there exists knowledge that we can apparently never know, because we are inextricably constrained by our limited capabilities.

For Kant, the world in itself, the noumenal world, is beyond us. Kant has an analogy of ‘irremovable spectacles’ of space and time through which our perception is enabled. Nevertheless, technological extensions to our sensory capacities, via scanning and computational augmentations to our reasoning and calculating may enable us to access knowledge exceeding that typically associated with human capability. Thus knowledge may become limitless, with a limitless increase of technologically-enhanced senses, with ever-more refined degrees of resolution, and perhaps even extending beyond limitations of human categories of understanding.

Colin Brookes, Loughborough

The quantity of knowledge that can be recorded is limited by the resources for data storage (including our brains), which, however large they become, are finite. A more philosophically interesting question is: are there kinds of knowledge denied to us?

One famous assertion that it is so is Wittgenstein’s “Whereof one cannot speak, thereof one must be silent.” However, this claim is in one sense false: there was a time when that ‘whereof one cannot speak’ was literally everything – and yet here we are speaking about all kinds of things. Better to say, whereof one cannot speak, one must devise new terminology.

One could argue that Wittgenstein’s claim was not meant to be taken so literally; but the reason why it isn’t literally true hints at why such attempts to define knowledge limits are presumptuous: we can’t know what new thought tools and media humanity will develop. Since emerging as a species, Homo sapiens has developed language, writing, logic and mathematics, diagrams, things to see the very large and very small, and more. All these qualitatively extend our ability to think far beyond what was possible for the earliest humans, who would not even have been aware of themselves as thinkers.

New tools are being invented all the time which may influence what knowledge can be recorded and transmitted. To give one example, in his essay ‘What Is It Like To Be A Bat?’, Thomas Nagel suggested that to objectively explain subjective experience, you would need to be able ‘to explain to a person blind from birth what it was like to see’. Fifty years later, researchers are working on techniques for direct brain stimulation for helping the visually impaired. Once these tools have reached maturity, you would not need to explain sight to a blind person, you could just give it to them. If there are limits to knowledge, the only way to know that is to reach them by continuing to step into the unknown and finding out what’s there.

Paul Western, Bath

Our knowledge is limited by many factors, including brain capacity, our sensory apparatus, the language we use, our experience, and our imagination. Our brain capacity affects the speed at which we can process information and recognize patterns, which limits our mathematical ability. Mental capacity also affects our linguistic ability, on which our ability to think critically depends. If we have no words to express a concept, we cannot think about that concept. George Orwell recognized this in 1984, in which the government tried to eliminate critical thinking by introducing ‘Newspeak’.

Our sensory apparatus determines our perceptions and perspectives, but our perspective can be extended using tools such as telescopes, microscopes, and spectrometers. Yet as Kant correctly stated, even using these instruments, we cannot know an object itself, as our perception of an object is always distinct way of representing an object. Even our ability to imagine is limited by our experience and existing thoughts, ideas, and concepts. New ideas usually arise from re-arranging the ‘known’ in different ways. For example the Romans knew about toasting bread, but had no conception of electricity, so that they could not imagine an electric toaster.

Using our sensory apparatus, tools, ability to detect and analyze patterns, linguistic ability, critical thinking and out imagination, we can scientifically increase our knowledge of our world. However our scientific knowledge is constrained by the above limitations, as well by the limits of technology.

Russell Berg, Manchester

There are two primary ways in which knowledge can be limited. The first involves information that is comprehensible to us but which there’s no way to obtain. For instance, we will almost certainly never know what Julius Caesar ate for breakfast on his seventh birthday. But if some oracle told us that it was bacon and eggs with soldiers (or a Caesar salad), that would be easily understood. There’s mathematical knowledge of a similar nature.

The knowledge of Caesar’s breakfast is in principle obtainable. However, it is completely impractical to obtain this knowledge. On the other hand, some knowledge is unobtainable even in principle. There are, for example, events occurring so far from us that, due to the expansion of the universe, no information from them can ever reach us. Light (or other information) emitted from these events will always be getting farther away from us.

The second type of limited knowledge concerns information we cannot even comprehend. Our minds are either too limited, or perhaps there is no explanation available. Possible examples of incomprehensible knowledge include: (1) The explanation of consciousness (as suggested by Colin McGinn); (2) The nature of free will; (3) The fundamental question of why there is something rather than nothing.

At least we are aware that these are problems. Perhaps there are even facts of which our minds cannot comprehend any aspect. There is no way of knowing what these facts concern, much less the facts themselves: in which case I can perhaps be forgiven for not giving an example.

Richard Stanley, Coral Gables, Florida

Because knowledge accumulates over the totality of subjects over time, one is inclined to say that knowledge is unlimited. I will give a simple approach to substantiate this statement. Consider the simple statement: ‘S knows that p’ . Then: if ‘S knows that p’ (e.g. ‘Laura knows that it rains’) then S knows one proposition – hence has limited knowledge. The first extension of this statement is to replace p with a finite set of propositions, (p1, . . ., pk). If ‘S knows that (p1, . . ., pk)’, then S knows a set of propositions, hence has limited knowledge.

Now suppose we also have a finite (n) set of subjects, (S1,. . . Si,. . . ,Sn). Then: if ‘Each subject Si knows a specific finite set of propositions’, we can say that (S1,. . . Si,. . . ,Sn) knows the union of these finite sets of propositions, or that ‘(S1,. . . Si,. . . ,Sn) knows that (p1, . . ., pm)’ , that is, knows a finite set of propositions, hence has limited knowledge, even though this is greater than for any one individual. Of course this group knowledge can be very large. Many subjects together know many things. We can also say that the totality of subjects knows the union of these specific unions of propositions. Or, using h as the number of members in this union, then ‘The totality of subjects knows that (p1, . . ., ph)’. This knowledge is still limited, because h is not infinity. Yet the totality of subjects existing at any moment in time (the world) knows a huge number of things. But by absorbing the sets of propositions from all previous totalities and all totalities to come, the number of propositions arguably approaches infinity. Totalities change over time, propositions are accumulate over time. Therefore we have, now and in the future, unlimited knowledge.

Teije Euverman, Rotterdam

The limits of knowledge aren’t difficult to ascertain: all that we require is a precision on what the word knowledge signifies. Knowledge can be defined as ‘information or data received by the conscious mind’. Information without a recipient is simply part of the stuff that makes up our cosmos; alongside energy and matter, it is fundamental. But once the sentient mind apprehends information, it is transformed into knowledge: data that can be used creatively. The limits of knowledge are therefore tied into the limits of mind and the limits of information. But we are unable to say where the limits of mind may lie. All we might say is that there should be an upper limit to what a human mind can comprehend – and it is here that we may then draw the limit of knowledge. But information also has an upper limit: all that exists, across space and time. Once this is fully reflected by a mind, knowledge has come to its completion. So knowledge comes to its end in omniscience – the apprehension of all that exists. Yet whether there was, is, or ever will be a mind that can have all knowledge is suspect. Some of the more speculative physicists have argued that this may be realised as artificial intelligence advances to its maximum and harnesses the energy of galaxies. Such ideas appear in the science fiction of Arthur C. Clarke and Olaf Stapledon. Many religions have proposed God alone (usually a God who transcends the Universe) as being omniscient.

It is interesting to consider whether the apprehension of such a God of those aspects of reality that transcend our cosmos would be knowledge in the sense we’ve defined it. It provokes us to also consider what information and mind might themselves be based on. But with God, language itself starts to become suspect; these are just some indications of the frontiers of what we might ask.

Anthony A. MacIsaac, Paris

Knowledge will reach its limits when there is one last question to ask and someone to ask it.

Isaac Asimov provided a better way of responding to this question than I could, in his short story, ‘The Last Question’ (1956). The story concerns the question, ‘What will happen when entropy ends the last star in the cosmos?’ A computer called Multivac is asked it, but, unfortunately its answer is, “There is, as yet, insufficient data for a meaningful answer.” Over the next billions of years, Multivac evolves in hyperspace, but always supplying the same answer to the question. As the last star dims, the last intelligent life – an ethereal entity – asks again, and receives the same response. The entity then joins with Multivac, which has gathered all the knowledge gained from trillions of years in the cosmos. When the last star dies, Multivac still has this one answer to provide. Eventually, after aeons of thought, Multivac finds the answer. With no one to report the answer to, Multivac decides to demonstrate; and so says, “Let there be light!”

As long as one question that has never been answered exists and there is an entity that can ask the question, the limits of knowledge have not been reached. I think the very last question would have to be: Can we understand how all the pieces of the jigsaw fit together? I therefore propose the limits of knowledge are reached when everything is known and understood. Or would it be? And would the answer to that final question tell us why it was all created in the first place?

Richard Tod, Desborough, Northants

Some people think that language determines the limits of knowledge, yet it merely describes what we know rather than limits it, and humans have always had the facility to create new language to depict new knowledge.

There are many types of knowledge, but I’m going to restrict myself to knowledge of the natural world. The ancient Greeks were possibly the first to intuit that the natural world had its own code. The Pythagoreans appreciated that musical pitch had a mathematical relationship, and that some geometrical figures contained numerical ratios. They made the giant conceptual leap that this could possibly be a key to understanding the Cosmos itself.

Jump forward two millennia, and their insight has borne more fruit than they could possibly have imagined. Richard Feynman made the following observation about mathematics in The Character of Physical Law: “Physicists cannot make a conversation in any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form.”

Meanwhile, the twentieth century logician Kurt Gödel proved that in any self-consistent, axiom-based, formal mathematical system, there will always be mathematical truths that can’t be proved true using that system. However, they potentially can be proved if one expands the axioms of the system. This infers that there is no limit to mathematical truths.

Alonso Church’s ‘paradox of unknowability’ states, “unless you know it all, there will always be truths that are by their very nature unknowable.” This applies to the physical universe itself. Specifically, since the vast majority of the Universe is unobservable, and possibly infinite in extent, most of it will remain forever unknowable. Given that the limits of knowledge are either infinite or unknowable in both the mathematical and physical worlds, then those limits are like a horizon that retreats as we advance towards it.

Paul P. Mealing, Melbourne

Knowledge has been defined as ‘justified true belief’. This is a philosopher’s view of knowledge. I would like to consider what we know and don’t know in a more basic, practical way.

First, we can have no real knowledge of the future. We may make informed guesses, even accurate predictions, but this cannot be counted as knowledge. Equally, although we know the broad sweep of the past, the everyday detail is largely unknown to us. We have reports of things that happened, were done and said, but this covers an infinitesimal part of the history of the world. Even the fine detail of what is happening around us today is largely unknown to us. In sum, our episodic knowledge is very limited.

It is more fruitful to consider what the limits are to what we know about the nature of reality, the physical laws which underpin it, and the entities that play a part in it. These are known unknowns that stand at the current limits of our knowledge. Some of the theories in need of more supporting evidence or explanation are dark matter, dark energy, string theory, quantum gravity, the multiverse, consciousness… Will full accounts of them be for ever beyond our reach? In some cases, there may be practical difficulties which mean that we will never obtain conclusive experimental evidence. Strings, if they exist, are incredibly small; Brian Greene said that a string is to an atom as a tree is to the universe. Thomas Hertog further said that “You would need a particle accelerator as large as the solar system to probe scales that small.” Yet, all hope is not lost: the LIGO detector was able to detect a change of one-thousandth the size of a proton in its 4-kilometre length caused by a gravitational wave! And AI with quantum computing may make lighter work of finding evidence and explanations. So will we eventually know everything there is to be known, or will there be an unfolding succession of unknowns, going on for ever like the extension of pi? Fortunately, we have five billion or so years before the Earth is engulfed by the Sun to find the answers.

Michael Brake, Epsom, Surrey

Next Question of the Month

The next question is: How Can We Achieve World Peace? Please give and justify your answer in less than 400 words. The prize is a semi-random book from our book mountain. Email the Editor. Subject lines should be marked ‘Question of the Month’, and must be received by 10 February 2024. If you want a chance of getting a book, please include your physical address.

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