- Computational Philosophy: not philosophy of computation, but philosophy using computational tools and methods.
- Mathematical Systems Optimization: for social, biological, and cosmic challenges: (This would probably be mainly theoretical now, but could be applied post-singularity).
- Virtual Sociology: how to manage a society that lives inside a simulated universe. Especially when the people in the simulated universe might be able to manipulate their reality at will. For instance: how do we prevent non-productive forms of hedonism + antisocial behavior from predominating in such a society?
- Neuronanotechnology. Not enough people focus on applying bionanotechnology in neuroengineering, I think that having neuronanotechnology as a named field could help bring more people into the area.
- Neuropoetry: kind of like cyberpoetry, but even more interactive. Example: you’re reading a poem and your brain is being stimulated at the same time to shape the experience in a unique, artistically meaningful way.
- Non-Invasive Cellular Connectome Mapping: methods and tools for in vivo imaging of neurons, synapses, and dendrites.
- Bioarchitecture: going beyond bionic architecture to actually incorporate biology and biotechnology into the structures of buildings. Example: growable buildings and partially growable buildings.
- Neuroaesthetics: new flavors of artistic meaning emerge from considering both the observer’s brain and the artwork being observed as part of a unified system.
- Physical Affective Neuroscience: attempting to quantify the exact physico-informatic structures that give rise to emotional qualia. The central goal of the field is to formally explain what arrangements of matter and energy states will lead to “positive” qualia and distinguish them from those that lead to “negative” qualia.
- Objective Ethics via Utilitarianism, Integrated Information Theory, and Panpsychism: research into an objective framework for morality. If an action will increase the net “positive” qualia across spacetime (as understood by panpsychic integrated information theory), the action is morally “good.” However, since we cannot measure qualia over all spacetime, we must establish a set of rigorous, probabilistic methods for estimating what will have “good” and “bad” effects.
- Theoretical Engineering: detailed proposals for engineering systems that are beyond current technology, but may be possible in the future. For instance, we don’t have a practical way to develop wormhole-based technology yet, but we have some theoretical basis for explaining wormholes mathematically. We might propose a method for developing ways to use wormholes in a technological fashion.
- Entomological Cybernetics: engineering-oriented study of control systems and communications within insect physiology and social organizations. Application of this research to develop artificial systems that mimic insects and cyborg insect systems (so long as these systems are demonstrated to be important enough that any harm done to the insects is outweighed by their benefit to other organisms; this means that the cyborg insects should not be used as toys).
- Human ecology: the study of humans from an ecological perspective. Care should be taken to avoid biases from harming the objectivity of this research (i.e. social Darwinism should be avoided).
- Ecoengineering: the engineering of ecological systems to maximize “good” qualia and minimize “bad” qualia. Computational ecoengineering may serve as a test ground for ecoengineering techniques.
- Dynamical Systems and Genius: there is a hotly debated correlation between genius and insanity. This field would study that relationship from the perspective of dynamical systems (stability, fixed points, bifurcations, etc.)
- Literary Exploration of Genius, Love, Risk, Imagination, and Madness: a literary body of work that deals with exploring themes that interrelate genius, love, risk, imagination, and madness. Also, the analysis of that body of literature.
- Romantic Rationalism: the development of a rational framework that does not ignore emotion as part of the human experience and recognizes the intrinsic value in “positive” emotional qualia. In fact, this form of rationalism would revolve around emotion while still using data, objectivity, and logic to help maximize the net gain in “positive” qualia across the universe.
When considering the fundamental substance of the universe, I am inclined to propose an information-based description. All physics may arise as a consequence of informatic processes. To unify this description with the experience of consciousness, I suggest that information and consciousness are synonymous. This would support panpsychism, the idea that “everything has at least some level of consciousness.”
To explain more completely, consider a rock. Any rock encodes information in its crystal structure. When sunlight hits the rock, energy is transferred into the atoms on the stone’s surface. From there, the energy propagates through the rock as heat. The pattern of thermal diffusion is dependent on factors such as the distribution of temperature in the rock at t=0 (immediately prior to contact with the sunlight), the relative densities and compositions of different parts of the rock, and the types of impurities present in the rock. As the sunlight shifts away from the rock, leaving it in shadow once again, heat begins to radiate back out. Depending on the processing inside the rock, the heat will radiate from different patches on its surface at different rates.
Now consider a human. When light contacts the human’s retina, a signal is transduced by opsin proteins and a cis-trans isomerization of the cofactor retinal. After several more steps, a signal encoding the pattern of light on the retina is transferred into the brain, where it is processed by an elaborate series of excitatory and inhibitory neural interactions. These neural processes take into account the individual’s past experiences, other sensory information, and more. The data is repeatedly transformed until it yields instructions for a motor response, perhaps turning the head away or blinking.
The rock and the human are similar in that they both are subsystems of the universe that take in data, transform it depending on internal structures, and generate some output. Of course, the rock does not experience the world in the same way that humans or even insects do. The rock’s experience is far more primitive. Compared to most biological organisms, rocks possess poor memories. The rock can store some hazy memories in its distribution of residual thermal energy from a previous encounter with heat, but these data are highly disorganized and difficult to retrieve in a form that resembles the original heat stimuli. Consequently, a rock probably lives “in the moment” and does not reflect upon its past experiences. Perhaps the stone experiences a fuzzy, often randomly changing, procession of sensations and mild swells of emotion, never really pausing to consider their implications.
By comparison, a human will experience more directed responses to specific stimuli. If a human sees someone she knows, some brain regions will be predictably activated. However, the human brain’s output is dependent on all current sensory information as well as its state at time t, leading to a colossal space of possible responses to an individual stimulus. The brain’s structure evolves over time as experiences accumulate, leading to variable responses even given identical sensory data. Unlike the rock, humans recall past events and so construct a continuous temporal context. With this context, humans can reflect upon their own experiences as well as predict future events.
Given these parallels between biological and non-biological information processing, I suggest that physical panpsychism may represent an accurate description of reality. This could provide a generalizable path to the neural correlates of consciousness, in which specific patterns of information are synonymous with specific conscious experiences. For instance, stable positive feedback loops might be involved in positive emotions like curiosity, excitement, and love. Of course, the human brain’s vast meshwork of data-transforming pathways gives rise to far more nuanced types of curiosity, excitement, and love than could be generated by an individual positive feedback loop. However, I would postulate that stable positive feedback loops could form the backbone for more complicated sentiments. It should be noted that some positive feedback loops can give rise to negative emotions (i.e. as in OCD). In these cases, the positive feedback might be coupled to other patterns of information which possess intrinsically unpleasant properties, overriding the intrinsic goodness of the loop. Another item to note is that positive feedback loops might only retain their pleasantness for as long as they are stable. If the loop can no longer reproduce or propagate its pattern (such as during habituation processes), the positive emotions may begin to fade. With this method of understanding consciousness, I argue that information structures may correspond to emotions in a quantifiable manner.
Topological Spaces and Topologies
Given a set X and a collection of its subsets τ (this τ is a topology on X), a topological space on X is denoted by (X,τ) and follows the rules below.
- X and the empty set are contained in τ.
- Any union of subsets in τ is contained in τ. This can be an infinite number of unions.
- Any finite intersection of subsets in τ is contained in τ.
- Given a topological space (X,τ) containing an element A, the neighborhood NA is any subset of X including A such that NA contains an open set of τ.
- Often, neighborhoods are defined around a point p in X. In these cases, the neighborhood Np is any subset of X including p such that Np contains an open set of τ.
- To visualize neighborhoods, consider a subset V of a plane X. The subset V contains a point p. If an arbitrarily small disk around p fits inside V, then V is a neighborhood of p.
- Note that, for a given set S and any point p on the boundary of the set, that set cannot be a neighborhood of p.
Open Sets and Closed Sets
- In practice, open sets do not include their boundaries and closed sets do include their boundaries. However, more complicated definitions exist.
- The union of any number of open sets is itself open.
- The intersection of a finite number of open sets is itself open.
- The complement of an open set relative to the rest of the space is a closed set.
- Some sets may be both open and closed. Examples of such sets are the given entire topological space X and the empty set.
Closure, Interior, and Boundary of a Set
- Informally, the boundary of a subset S includes the points on its “outline.”
- Slightly more formally, the boundary includes the points in S for which no open ball will be entirely inside S.
- The interior of a subset S includes all points in S that are not part of its boundary.
- The closure of a subset S on a topology is defined as the union of the boundary points of S and the interior points of S.
- A basis B for a topology X is a family of subsets of X such that every open subset of X is the union of some members of B.
- The empty set is the union of any nonoverlapping elements of B.
- Any intersection of base elements is another base element.
- Many bases may generate the same topology.
- A common example of a basis is the set of all possible open balls (in 2D they actually are disks, though they are called balls no matter the dimensionality) that union to form a plane.
- Below, just some of the (infinite) open balls forming the basis for a topology consisting of a 2D shape are illustrated.
Continuous Functions between Topological Spaces: Homeomorphisms
- Functions between topological spaces generalize the notion of real and complex-valued mappings to any rule that assigns abstract objects in a domain to abstract objects in a codomain.
- Continuous functions between topological spaces are called homeomorphisms when the inverse of the given mapping is also continuous.
- Given a mapping from a topological space (X,τ) to a topological space (Y,τ), the mapping is continuous if and only if all open sets in the domain map to open sets in the codomain. That is, the preimage of any open set in the codomain must be open in the domain.
- Another property to note is that homeomorphisms are bijective (each element in the domain maps to exactly one element in the codomain).
- Given that (X,τ) and (Y,τ) are homeomorphic, any topological property (such as connectedness) of (X,τ) or of (Y,τ) must be shared by both (X,τ) and (Y,τ).
- If (X,τ) and (Y,τ) do not share even a single topological property, then (X,τ) and (Y,τ) are not homeomorphic.
- The common joke about how topologists consider donuts and coffee cups to be the same arises because those items are homeomorphic (if they are assumed to be topological spaces).
Hausdorff Spaces, Compactness, Paracompactness, and σ-Compactness
- Hausdorff spaces are topological spaces in which individual points have disjoint neighborhoods. Disjoint neighborhoods are neighborhoods that do not share any elements (the intersection of disjoint neighborhoods U and V is the null set).
- Note that the term “disjoint” can also refer to sets. Disjoint sets are sets that do not share any elements.
- Compact spaces have finite open covers. An open cover of a set X is a collection of sets that contains X as a subset. As such, a finite open cover involves a finite collection of sets containing X.
- Given a cover, a subcover of X is a subset of a cover that still contains X. The subcover of X is equivalent to the refinement of X.
- Paracompact topological spaces are spaces for which every open cover contains a locally finite open refinement. A collection of subsets is locally finite when each point in the space has a neighborhood that intersects a finite number of neighborhoods in the collection.
- σ-compact topological spaces are topological spaces which can be described as the union of countably many compact spaces.
- Note that a countable set is a set for which each element is associated with a unique natural number. More formally, countable sets have the same cardinality (number of elements) as a subset of the natural numbers. Countable sets may be countably finite or countably infinite. When countably infinite, each element can be counted one at a time, but the counting may never finish.
First-Countable and Second-Countable Spaces
- First-countable spaces are topological spaces for which every point is an element in a countable number of neighborhoods. For each point, this set of neighborhoods is called a neighborhood basis or a local basis.
- For example, consider a topological space on a set X consisting of four points x1, x2, x3, x4. (Assume that the set is equipped with a topology). Each point has a countable number of overlapping neighborhoods. In the figure, the neighborhoods which include point x4 are shown as an example.
- All metric spaces are first-countable since any point in a metric space will have a countable (though often countably infinite) number of overlapping neighborhoods. This comes from the fact that distances are strictly defined on metric spaces, so the number of open balls making up the basis for a metric space will correspond to some natural number.
- Second-countable spaces are topological spaces with a countable basis.
- Euclidean space is a second-countable space because the set of open balls forming its basis can be restricted to the set of open balls with rational radii and centers with rational coordinates.
- Every second-countable space is first-countable (but not every first-countable space is second-countable).
- Connected spaces are topological spaces that cannot be represented as a union of disjoint nonempty open subsets. That is, a connected topological space X cannot be divided into disjoint nonempty open subsets.
- For instance, the usual topology (described in the next section) on Rn is connected as any union of disjoint open subsets will always exclude at least part of the n-dimensional real numbers.
- Disconnected spaces are topological spaces which do not satisfy the definition of connected spaces.
- Path-connected spaces are topological spaces for which a path can be used to draw some curve from any point x to any point y in the given topological space X.
- A path is defined as a continuous function mapping from the unit interval [0,1] to the topological space X such that f(0) = x and f(1) = y.
- Every path-connected space is connected (but not every connected space is path-connected).
- Simply connected spaces are topological spaces which are path-connected and every path between every pair of points can be continuously transformed.
- Given a topological space V, it is said to be locally connected at a point x when every neighborhood of x contains a connected open neighborhood (a topological neighborhood that does not include its boundary). When V is locally connected at every point it contains, then V is called a locally connected topological space.
Essential Examples of Topological Spaces
- The standard (or usual) topology on Rn may be defined as the union of all possible open balls on n-dimensional Euclidean space.
- Given a set X, the discrete topology on X is a topological space in which each point forms an open set. Each of these open sets is called a singleton (a set with only one element).
- Given a set X, the indiscrete (or trivial) topology is a topological space in which the only open sets are X itself and the empty set. If X has multiple elements, the space cannot be equipped with a metric (any distance between elements must be zero).
- Given a topological space (X,τ) and a subset V of X, the subspace topology is the subset V equipped with the topology τ.
- Topological manifolds are a type of topological space which must satisfy the conditions of Hausdorffness, second-countability, and paracompactness.
- In addition, topological manifolds must be locally homeomorphic to Euclidean space, Rn.
- In order to be locally homeomorphic to Euclidean space, the formalisms below must hold.
- This means that, for every point x in a locally Euclidean topological space X, there exists an open set U such that x belongs to U and there exists a homeomorphism h from Rn to U. These homeomorphisms are called charts. The combination of charts which covers X is called an atlas.
- Below is a simplified representation of a manifold (only a few homeomorphisms are shown). In this case, the manifold is a 2-manifold, also called a surface.
- Manifolds cannot “cross themselves” unless special conditions are met (i.e. the Klein bottle is an exception). For instance, a 1-manifold cannot have curves which intersect and a 2-manifold cannot pass part of its surface through another part of its surface. The reason for this is that Euclidean space cannot be homeomorphic to a point of intersection since this gives multiple values for the same function and therefore is not defined as a function.
- However, as the dimension of a manifold increases, the manifold can cross itself when projected into lower dimensions while not actually crossing itself in its highest dimension. Consider that a surface may “fold over” to some degree without intersecting itself in R3. If projected into R2, then this manifold will appear to cross itself. But when looking back at the R3 depiction, it is clear that the manifold does not actually cross itself. The same holds true for higher dimensions.
- For an atlas, two charts can overlap on a manifold. The intersection of these two charts is an open set which maps to the same region of Euclidean space. Transition maps are composition functions f(g-1) = f∘g-1 and g(f-1) = g∘f-1 which map the open sets in R2 to the manifold (by the inverse function) and then map back to Euclidean space (by the function which describes the other homeomorphism). This is also called a coordinate transformation.
- If a manifold’s transition maps are all at least once differentiable (C1), then the manifold is called a differentiable manifold.
- If the transition maps are all infinitely differentiable (C∞), then the manifold is called a smooth manifold.
I have always enjoyed looking at images that display biological systems in terms of their functional components. But oftentimes, existing diagrams show too much detail, too little detail, the wrong types of detail, or confusing captions/explanations. This seems especially common in neuroscience. Inspired by this challenge, I put together the image below to explain the fundamentals of hippocampal circuitry in a clean, unified manner.
Some exciting/beautiful/inspiring topics to explore!
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