Week 8 (5/13,15): Digital Doubles

[Gio-Choi]

This week we consider the use of computation and data to create models or “digital doubles” of scientific and technological phenomena of interest in order to understand it, discover it or invent it. I first introduce a core example that coevolved with early, large-scale computation, the Monte Carlo method, which uses uncertainty about what we don’t know about phenomena, rather than just we know; then I introduce the “Unreasonable Effectiveness of Mathematics in Natural Science” by Nobel laureate Eugene Wigner about the underexamined appropriateness of math for science, an example of an early 19th Century algorithm being pickup and doing new scientific work in the computer era. Finally, I have you consider the inextricable relationship between climate science and climate modeling in Edward’s book. Why do scientists build computer models of natural (or social) phenomenon? What can scientists and engineers do with models that they cannot (or may not) be able to do with the phenomenon? How does one determine whether a model is “good enough” in its correspondence to the real work to learn from? What is the relationship between a model and a theory in science—are they the same thing? And what happens when two (or more) equally strong but distinct models emerge within a field—are there pressures for convergence?

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[suryachinnappa]

An adage attributed to scholar Alfred Korzybski is “the map is not the territory”. The phrase addresses a common logical fallacy, which is to reason as though an abstraction of a certain thing is that very thing. This fallacy naturally arises from the fact that in many scientific endeavors, we are in possession of models so isomorphic to that which they represent that it becomes easy to confuse that which something means and that which something represents. This week’s readings deal with the antecedent to the map-territory relation and its associated fallacy: the very fact that we are in possession of such accurate models is far from trivial.

One of the readings for this week, “On the Unreasonable Effectiveness of Mathematics in Natural Science”, by Richard Courant, established this idea very strongly, touching on the interrelations between mathematics and the natural sciences, particularly physics. Multiple angles towards the issue were worked out: on one hand, the fact that “laws” of nature must be discovered, and subsequently mapped onto mathematical constructs, highlights the fortune of scientists and knowledge-seekers at large in the fact that we happen to inhabit a world in which many natural phenomena can be modeled mathematically using rather simple concepts. Here, mathematics is positioned as an unchanging, upright pillar of rationality, built on solid foundations from which the laws of physics are given a framework and structure. On the other hand, this positioning is not without its demerits; Courant notes cases like the complex numbers, a concept which has no easily accessible natural isomorph yet which is still vigorously defended and cherished by mathematicians, which on first glance look like nothing but pure abstraction, but in fact reflect useful properties and relations from physics. In more modern times, the domain of mathematical physics bridges these two disciplines, and indeed, many mathematical concepts are directly inspired by physical reality. We realize that these concepts are nothing but our own human constructions; Courant notes that most mathematical work is done from such structures, not from the axioms of the mathematical system, but since our definitions of these structures can be placed rather arbitrarily, they end up existing often where they are most useful for applied mathematics. Thus we see that just as mathematics structures physics, physics informs mathematics; the relationship between the formal system and physical reality is a bidirectional one. A similar process can be understood with lexification and social constructs. Giving a word to something previously anonymous makes it instantly tangible, allowing it to participate in social reality; we can relatively easily see that we give words to things we perceive as parts of our reality, and in turn, words cause things to become parts of our reality.

To briefly touch on computation: Accelerating mathematics with computational tools has facilitated the realization of models to a greater degree than ever before. Monte Carlo simulations are one powerful example, but so are such things as spiking neural networks for the digital doubling of neuronal systems.

Discovery:
Reading the paper of the semantic tiling of the human cerebral cortex was very fascinating. The fact that we can use such computational methods to get a picture into some of the most abstract contents of our brains are an enticing promise; however, I am a little uncertain as to the strength of their methods. In the Methods section, they note that they created their feature space informed by a very large corpus of text. While the high dimensionality of their feature space meant that it likely did encode valuable information about neurally-embodied representations of semantic concepts, psycholinguistic research has shown that our own linguistic predictive capabilities do not fully mirror those than might be sourced from corpus statistics.

[aadarwal]

These readings were particularly interesting because while a lot of the stuff that we have read before are ideas that I have at least remotely read or thought about, a lot of this was completely new to me. For instance, the fact that we build computer models not merely because the real world is too vast, volatile, or radioactive to handle directly, but because simulation lets ignorance itself become an instrument of inquiry was something I was not aware of. Ulam’s Monte Carlo method turned random card-dealing into forecasts of neutron diffusion, proving that when equations buckle, randomness can sketch the contours of the unknown. Wigner, still puzzled that abstract mathematics keeps “biting back” with uncanny accuracy, reminds us that a model’s power often lies in how widely its math migrates and still pays empirical rent. Edwards shows climate science institutionalizing this logic: global climate is less an observed object than a vast machine of stitched-together data and general-circulation models that replay the past, fast-forward futures, and run counterfactuals policymakers can legislate against—models validated through calibration to history, blind hindcasts, and ensemble convergence. Here theory is the conceptual scaffold (energy balance, thermodynamics, and the model is its executable compromise. When rival models of equal statistical muscle coexist (early aerosol-cooling vs. CO₂-warming climates, or The Limits to Growth’s world dynamics alongside GCMs), convergence pressures arise from “data gravity,” policy demands for a single voice, and inter-comparison projects—yet diversity persists because each model’s errors cancel another’s blind spots. Pushing the idea into society, Kozlowski and Evans warn that large-language-model “stand-ins” may let us run cheap, infinite social experiments while quietly outsourcing our ignorance of genuine motives. The moral across domains is the same, which is that models are epistemic prostheses and useful, provisional, theory-laden sandboxes where we can ask questions without burning bombs, planets, or research budgets. Their adequacy is judged not by perfect mirroring but by asking questions about their nature and use cases. What this means then is that Wigner’s wonder combined with Monte Carlo’s dice keep multiple models talking, keep the randomness rolling, and dont allow a machine work truly in a non-noisy manner.

Discovery:
My discovery was more particularly revealing in the canvas assignment reading because I think that reading was the one that made me think the most. Particularly the reading for the understanding of predicting linguistic/ language words in the brain and mapping centers of word production was insightful because I have always believed that human freedom in some way could be related to the linguistic relation between the brain and some sort of non computational reliance on the imprecise language that we use (or maybe more in line with Penrose's logic, regardless same kind of idea maybe), and looking at the work that was done made me doubt some of the faith that I have in such a belief and I was tempted to maybe wonder if this freedom was maybe a result of the social actions we perform being indeterministic, and without a way to actually test that at the moment the study has definitely left that confused after this discovery.

[cavanbgon]

Throughout this course, we have observed the emergence of objects developed for the expressed purpose of having “ugly” or otherwise processes passed onto them. This was the first goal of calculating machines, however, variously they were formulated. As we edge towards mature computational systems in the modern context, this urge, which was by-and-large abandoned in favor of marginal improvements in the efficiency with which various desired operations may be accomplished, seems to reemerge, albeit in a more general sense.
Rather than seeking to “pass off” work viewed as somehow “inessential” to a more mature mind, we observe the emergence of a notion of a process up to what is “computationally essential”. The example of solitaire, an understanding of whose structure was fundamentally motivating for the development of Monte Carlo methods, is instructive. Defined as a game, Solitaire may be viewed as distinguished by one’s experience in it. This is to say that a game of Solitaire may be understood as such to encompass the place of its occasion, the enjoyment of the game’s operator, among other features. This is all without referencing the identities of any cards placed or turned over through the completion of a game. In the creation of a “digital double” (following Ulam’s conceptualization on which Monte Carlo methods act), one has decided to strip down the game of Solitaire to the ordered sequence of “moves” which occur on the game’s cards. With such an "essentialization" established, one can begin to use the tools of computation recently developed to collect masses of data about this invented entity. In the case of the Monte Carlo method, we wished to understand otherwise nearly intractable probabilistic features of arbitrarily complex systems (“the system” being understandable only as a particular abstraction of a prior phenomenon, the one which putatively one actually wishes to gain insight into).
With such a general technique established, many systems initially lacking a well-understood mechanism for their operation become, in some sense, far more tractable. Returning again to early applications of the Monte Carlo method, we see an explosion of new work in the modeling of particle-particle interactions as occur in nuclear reactions. Similarly, the structure of certain biological “patterns” previously identified with discrete mathematical patterns may, if reducing these phenomena to their singular mathematical phenomena placed within a quasi-evolutionary process, receive an ontogeny.  In an even more striking case, we see a concerted effort to develop a unified collection of atmospheric data previously confined to small localities extended to the entire globe, producing in turn the first strong notion of a simultaneous, interacting global climate.
Though these efforts have obviously been essential to the depth of our contemporary understanding of many natural and social phenomena, the act of creating an abstracted double whose operations are studied presents some potentially deep issues. Even taking Wigner’s observation of the arbitrary power of mathematics for achieving insight into physical problems as a kind of postulate, such optimism can never be relevantly extended to the choice of an abstracted computational model for understanding a model of physical problems. Without a deep sensitivity to the properties of the model one has developed to be representative, it is exceptionally easy to achieve exceptionally mathematically sophisticated, statistically rigorous tautologies.
Discovery:
Reading the article on Inverse Molecular Designs, I felt kind of at a loss. On the one hand, I’m generally positive about the possibilities of deep learning systems to effectively predict various new structures as they may emerge in broad areas. At the same time, intuitively, it would seem that such a process, fully automated, without the intercession of strong human intuition to guide the system, is incapable of producing an output that can give information more than tautological. The complexity of the system must be proportional to the strength of intuition with which a system is first formalized. In the cases of language models and proteins, the terms by which we evaluate these are sufficiently within human intuition and governed by sufficiently strong human-derived principles that the architecture of achieved constructions should be at least locally interpretable. Without this, stating that closeness of vectors within a higher-dimensional metric space is equivalent to “similarity” becomes a tautology.

[evancholerton]

This week’s readings collectively see models as epistemic prostheses: tools that deliberately embrace and structure our ignorance to learn about complex phenomena. This idea is woven through Monte Carlo’s leap from games of chance to neutron diffusion, Wigner’s “miracles” of math biting back in physics, and Edwards’s “vast machine” of climate data and models.

Let's start at Los Alamos in 1946, where Stan Ulam, recovering from illness, wondered whether instead of combinatorial gymnastics he might “lay [Canfield solitaire] out say one hundred times and simply observe and count the number of successful plays”. In that moment, randomness became a scientific instrument, not just noise. Von Neumann saw that statistical sampling on the new computers could solve otherwise intractable differential equations, estimating neutron-multiplication rates by tracing individual “chains of events” rather than wrestling huge integrals. The Monte Carlo method thus pipelines uncertainty into a structured experiment.

Wigner puts this in philosophical relief. He marvels that concepts like complex numbers nevertheless prove “close to being a necessity in the formulation of the laws of quantum mechanics”. More broadly, he calls it a “miracle” that mathematics, developed independently of physics, so accurately “bites back” with testable predictions. Here again, the unknowable (quantum behavior, heavy-atom spectra) yields to abstract structures we didn’t empirically derive but which nonetheless map reality.

Edwards’s A Vast Machine brings these ideas into climate science. He shows that only “about ten percent of the data used by global weather prediction models originate in actual instrument readings; the remaining ninety percent are synthesized by… ‘4-dimensional data-assimilation’ models”. Climate reanalysis goes further, running decades of historical observations through a single, “frozen” model to produce a consistent global dataset. Yet this generates new biases, so scientists must perform infrastructural inversion probing every station’s history, “linking changes in the data record to social and technical changes in the infrastructure that created it”.

Across these domains, models do more than simulate: they shape both our data and our theories. They turn ignorance into an experimental variable. This raises questions: when does a model stop “double”-ing reality and start defining it? How should we balance multiple models’ blind spots? And as AI systems increasingly serve as our digital doubles, what new pressures will converge or diversify these synthetic worlds?

Discovery
I was struck by Ulam’s simple but profound insight that “ignorance itself” can be wielded: laying out solitaire hundreds of times rather than chasing unwieldy combinatorics. It revealed that randomness isn’t merely tolerated, it’s a source of knowledge. Today, AI training techniques like dropout or random forest ensembles echo this heritage, leveraging uncertainty to build more robust inference. This lineage reminds me that in both computation and society, we gain power not by pretending certainty, but by crafting tools that can ask “what if?” across the unknowable.

[ucachyut]

Original Materials

Computer models allow scientists to examine phenomena that are difficult, dangerous, or impractical to interact with directly. For instance, scientists can simulate black holes or early universe conditions well before those systems are observable in any empirical sense. These models offer a way to explore, test, and visualize abstract theories and are often the first step toward verifying hypotheses that could otherwise remain speculative.

Weather forecasting is a particularly useful example. Unlike the weather itself, which can only be observed as it happens, computer models allow scientists to make predictions about future states of the atmosphere. This predictive capacity is what separates modeling from passive observation. In general, computer models extend science into realms that are either inaccessible or temporally disjoint from immediate observation—both future events and theoretical possibilities.

The Monte Carlo method exemplifies how uncertainty itself can become a tool. Rather than trying to eliminate variability, Monte Carlo methods use random sampling to systematically explore complex systems. Originally developed for nuclear simulations, this approach models phenomena that defy deterministic solutions. It reframes ignorance not as an obstacle, but as a productive condition—one that allows researchers to generate probabilistic insights from limited information.

A model is considered "good enough" when it successfully accounts for the full range of observed phenomena within a given scope. For example, Newtonian mechanics continues to be useful because it accurately models a large class of physical behaviors, even though it breaks down under relativistic or quantum conditions. The performance of a model, then, is not absolute but contingent on the domain of application. As observation improves, models are updated accordingly, and their reliability is tested against broader and more refined data.

A model and a theory are often treated as distinct, but they are essentially equivalent in structure and practice. A model provides a general mathematical method to predict phenomena based on observations, while a theory seeks to explain why those observations occur as they do. However, most theories only offer explanatory power after developing a working model of how the system behaves. In that sense, models and theories are two sides of the same process—models guide the construction of theory, and theories give models conceptual justification. In modern science, theories typically contain computational models as central components for prediction and explanation.

When multiple models coexist, pressures for convergence often arise—not only from the scientific community's desire for coherence but also from external needs such as policy-making. As Edwards argues, climate models are not merely scientific tools; they are embedded in political contexts that demand consensus. Disagreements over models often extend beyond epistemology into questions of governance and legitimacy.

Modeling social systems adds a further layer of complexity. Human agents cannot be reduced to fixed rules or rational optimization without loss of nuance. Kozlowski and Evans note that AI-driven simulations of social behavior can be analytically useful but risk oversimplifying real-world dynamics, raising ethical questions about representation and validity.

Discovery:
In reading Paul Edwards’ A Vast Machine, I was struck by the idea that climate models are not just computational artifacts, but also sociotechnical systems built on layers of historical observation, standardization, and global coordination. The claim that “you can’t do climate science without models” reflects not only the technical complexity of the climate system but also the political and epistemic infrastructures required to stabilize shared datasets and methods. This helped me understand that models are never purely mathematical; they are shaped by institutional choices, funding priorities, and international norms.

Fermi's Calculations Nuclear reactions are an example of phenomenon that are difficult to observe but easier to compute.

[benkim04]

Scientists build computer models of natural and social phenomena in an attempt to grapple with its complexity. That is, to simulate systems beyond the reach of direct observation or experimentation in order to hopefully gain a deeper understanding of risks, dynamics, and specific phenomena than direct measurement or observation can achieve. These models serve as “digital doubles,” standing in for real-world systems and allowing researchers to explore hypothetical conditions, test interventions, and generate data where empirical collection is infeasible.
The Monte Carlo method exemplifies the power of computation to work with uncertainty. Rather than trying to eliminate randomness, it attempts to use a model with probabilistic outcomes in complex systems. It shows not just what we know but gives structure to what we don’t know, providing a methodological foundation for inference under uncertainty. We now understand probabilistic modeling to be hugely important in the modern day.
“The Unreasonable Effectiveness of Mathematics in the Natural Sciences” begins to lay the groundwork of how we use mathematics (and models) to understand natural phenomena. Wigner raises questions about whether mathematics is discovered or invented, and why it should align so well with empirical phenomena. His reflections underscore the idea that models often precede discovery, meaning computational techniques from earlier centuries have acquired new potency in the computer age due to the newfound ability of high-speed calculation and otherwise.
In fields like climate science, modeling is not just a tool for understanding climate. It is critical for knowledge production within climate. That is, we do not have a second Earth to experiment on. Climate models, built from physical theories and observational data, allow scientists to simulate the planet’s response to varying greenhouse gas levels, ocean currents, or solar cycles. We can’t experiment with varying greenhouse gas levels or ocean currents in any practical manner, so all knowledge must be extracted through models and simulations.
But how do we know a model is “good enough”? Although being accurate is always important, a model’s utility is not necessarily in its absolute accuracy but in its ability to produce explanatory and predictive insights within defined parameters. Scientists validate models through calibration with empirical data, sensitivity testing, and by comparing outputs with real-world outcomes. A model is never a perfect replica; it is a tool. Its adequacy depends on its purpose. If we aim for explanation or prediction, more accuracy could be desired. If it is for drafting policy, perhaps the model does not to be quite as accurate, and just provide a general idea about the system.
When multiple, equally strong models coexist, the scientific community faces a largely unavoidable problem. Convergence may be driven by empirical success, methodological consensus, or institutional pressures. The multiple models do inevitably compete, which likely drives them to improve individually, or one being crowded out by the other. Either is most probably beneficial, as more accurate models will likely come out of such competition.

Discovery:
When reading “Simulating Subjects: The Promise and Peril of AI Stand-ins for Social Agents and Interactions”, I was struck by the prospect of creating models not of traditional natural phenomena, but instead of human beings. That is, it seems obvious to me where it would be possible to experiment in the real world with things like climate. However, human subjects are not only available, but at least in my mind are irreplaceable. That is, the story of each individual, how they grew up, and the people they have meet are all parts of them, and thus to model them would be fruitless. Further, to base research and real scientific inquiry on that seems absurd to me upon initial inspection. Still, it is obvious that it could be helpful in certain situations.

[Alex7692]

Original material

Scientists build computer models of real-world events to, simply, accomplishing research tasks that are implausible to conduct physically, viz. due to structural complexity, cost and various other stakes. The models are manipulable with high precision. The experiments, systematically replicable. They are nonetheless "models" of reality (and by that, already assuming a certain correspondence theory of truth). In the case of the Monte Carlo method, it would seem like the computational algorithm based on random sampling is the only viable method in replacement traditional analytical approaches –––– although bound by the curse of dimensionality. Whereas this may less be of an issue for the investigations of natural phenomena (cf. the "unreasonable effectiveness of mathematics"; that is, we seem to already possess the language to regiment the laws of nature), the discrepancies may be less negligible when it comes to sociological / humanistic analyses.

To what extent could we then assert whether a model is "good enough" in its virtualization of reality? Consider Schopf and Raup's seminal work on stochastic paleontology. His illustrates how macroevolutionary patterns can be approximated as random walks. Supposing Raup's right, it would seem to follow that the law of evolution is governed by statistical principles. It would therefore radically undermine the tacit assumption of determinism in paleontological research (which, for a long time after the modern synthesis, has troubled the paleontologists). The results met objections from Stephen Jay Gould, who argued trenchantly for the model's negligence over other particular, contingent factors beyond strictly Darwinian principles that had contributed to macroevolution. The model cannot explain patterns as such as mass species extinction, particular allometric patterns, and the punctuated equilibria from a "literal reading" of fossil records (Gould was to some extent influenced by Kuhn and the idea of "paradigm shift" in developing an account for these phenomena). There are broad shifts in the tempo of macroevolution that are irreducible to the gradualism of microevolution. To place this intellectual debate in the context of our discussion: it is totally justifiable for us to model phenomena as complex and irreproducible as evolution of prehistoric species. But insofar as it reveals how a random walk is "close enough" to the actual evolutionary patterns for us to conjecture its nature as fundamentally stochastic (and as such, the model itself becomes the "theory"), it's almost certain that the approximations are infinitely delayed from matching reality. And that gap is precisely what Gould sought to hold austere.

One curious point that I'm considering is how, for instance, a pragmatist (or interventionist) philosophy of science would account for digital doubles. Ian Hacking famously argued that the reason why scientists take physical entities (e.g. electrons) to be real is because we possess the ability to manipulate them. We act rather than observe. Are we acting similarly with digital doubles?

Exploration

This is an old material but I have had the chance to use it yet. The "conversion machine", basically (I'm afraid I could not remember who the author was). This would seem like a perfect scenario in which a digital double is needed.

I've also read the article on semantic maps. While the researchers have employed a data-driven approach and developed a statistical model based on raw data, the results are ultimately compared to fMRI data for corroboration. There is a sense in which virtualization is used as a way to probe at reality, viz. reflect back onto real data sets.

[Laurelli7]

Original material

Scientists build computer models of natural or social phenomena when real-world experiments or direct analysis are impractical or impossible. Wigner’s observation that mathematics somehow fits nature “bordering on the mysterious” suggests that formal models often capture essential structures of reality. For example, climate researchers saw Earth as “a single, interconnected whole” only after satellite data and computing enabled a global perspective. In social science, AI models now offer “stand-ins” for human subjects that can be run in large numbers – an experimenter can simulate thousands of trials with virtual agents in ways that would be impossible or unethical with real people. In summary, models let scientists manipulate systems (from atoms to societies) in silico to ask “what-if” questions that experiments or observations alone cannot answer.
With models one can conduct computational experiments that are impossible on the actual system. For instance, Raup’s shell model allows exploring every conceivable coiled shell shape in a three-dimensional “morphospace”, even forms never seen in nature. By contrast, one cannot physically create those shells or rewind evolution. In physics, a Monte Carlo model can simulate countless random particle histories to estimate a quantity, whereas an actual reactor experiment might be too dangerous or expensive. AI social simulations let researchers test how group dynamics change under controlled variations (like personality prompts) at scale, something that would be prohibitive with human subjects alone. Models also allow scaling time or size: climate models can fast-forward decades of change, and ecosystem models can integrate centuries of evolution or thousands of years of data, achieving insights not reachable by short-term measurement.
Determining when a model is “good enough” depends on context. In statistical simulations, one often looks for convergence and acceptable error margins. Metropolis and Ulam noted that their Monte Carlo estimates become reliable “with great probability” once many trials are run. More broadly, a useful model should reproduce known data and make valid predictions. Edwards stresses that modern models are fitted and tested against empirical data: climate models are “filled with data” so that their outputs align with observations. In practice, scientists calibrate models and compare them to experiments or field data (e.g. hindcasts in weather prediction) to validate them. Goodness may also be judged by how well a model answers the questions at hand: if the model captures the key mechanisms and passes basic checks, it can be “good enough” for exploring new scenarios. Wigner’s insight reminds us that even a model that fits current data is not guaranteed unique, so scientists must remain cautious and seek further tests.
A model is not the same thing as a theory, though they are related. A theory is a general explanation (often mathematical) of how a system works; a model is a concrete instantiation of that theory (usually computerized) often augmented by simplifications, parameters, and data. Edwards points out that models are “not pure theories” – they incorporate empirical inputs to tie theory to reality. For example, the Navier-Stokes equations (theory) underlie climate models, but the actual code includes many parameterizations based on data. Similarly, Monte Carlo is a method built on probability theory: the theory defines the rules, and the Monte Carlo model executes them in a simulated experiment. Thus models operationalize theories and make them testable. They can incorporate empirical adjustments in ways that pure theory does not, but unlike theory they can be directly run to produce numerical predictions.
When two equally strong but distinct models arise, scientists must seek criteria to choose or reconcile them. Wigner’s discussion of possible alternative theories is relevant: he suggests we have “no definite evidence that there is no such theory” that explains phenomena just as well. In practice, this means competing models may coexist until evidence breaks the tie. Often, multiple models encourage convergence through synthesis or averaging: for instance, climate science uses ensembles of different models to span uncertainty. In other fields, pressure for simplicity or parsimony can favor unification. On the other hand, distinct models can highlight different assumptions and drive new experiments to distinguish them. Ultimately, science tends to resolve such conflicts by gathering more data, refining theory, or demonstrating one model’s wider applicability. As Edwards notes for climate, the simplistic “models vs. data” debate was replaced by more sophisticated integration – similarly, disparate models often lead to efforts that unify insights or reveal when one model’s domain of validity ends.

Discovery:
However, I believe such simulations using digital doubles can cause real harm, especially as they become more advanced and difficult to distinguish from authentic human interactions. These simulations often rely on data that individuals have unknowingly or passively contributed online—through social media posts, emails, search histories, and digital footprints. Tech companies aggregate and process this information to build AI models capable of mimicking human behavior with increasing accuracy. While this may appear innovative, it raises serious ethical and identity concerns.
One major issue is consent. People rarely realize that bits of their personality—language patterns, emotional expressions, or even decision-making tendencies—are being used to create virtual replicas. These AI-generated "doubles" can act, respond, and communicate in ways that resemble real users, potentially without their knowledge or control. This leads to troubling questions: if an AI sends a message or takes an action on behalf of someone, how do others know it’s not the person themselves? What happens when simulated agents start interacting in public spaces like forums or social media without disclosure?
The boundaries between real and simulated personas become blurred, introducing the risk of identity confusion or manipulation. A hyper-realistic digital simulation could be mistaken for a real person—intentionally or not—raising concerns about misinformation, impersonation, and trust in online communication. As simulations improve, we will need robust safeguards to ensure transparency and to protect individuals from having their identities digitally co-opted.

In addition, from the trip to Stephen Stigler’s personal library collection, I discovered the first of few maps created by collecting real geographical data. Maps are normal in our daily life but it is a early example simulation from my perspective.

[peterzhou-cs]

So, what is mathematics, really? I’ve been thinking about this, and I keep coming back to the idea that math isn’t just one thing—it’s kind of two things at once. On one hand, you have the theoretical side, the kind the ancient Greeks were into. They weren’t doing math to build things or fix practical problems. They were exploring abstract ideas—like whether you could square a circle using just a ruler and compass. It wasn’t about doing something in real life—you could easily fake it with some measurements—it was about what was possible in theory, with perfect logic. That’s one whole tradition: formal, abstract, not meant to be useful.

Then there’s the other side: math that’s made for a reason, to solve real-world problems. Think calculating taxes, measuring land, designing machines. In this tradition, math is a tool. And tools are supposed to be useful. If we design something for a specific purpose, then it’s not surprising when it works.

But here’s the strange thing: real mathematics today lives somewhere between these two. Not all abstract math ends up being useless, and not all applied math is purely utilitarian. A lot of abstract ideas somehow end up becoming useful in surprising places. That’s where the mystery comes in.

Take pi, for example. It starts off as this basic circle thing—the ratio of a circle’s circumference to its diameter. Simple, right? But then it turns up everywhere: in statistics, physics, even gravitational constants (which, depending on how we define our units, and pi shows up because of the pendulum). Sure, sometimes that’s just convention—but it still feels like more than coincidence.

What is how math gets reused. A formula or constant doesn’t just work once; it shows up again and again, in completely different domains. That doesn’t happen with things we design for just one task. But maybe it makes sense if we think of math like medicine. A good medicine works because it activates a mechanism in the body—not just for one illness, but maybe for others too. You don’t see the mechanism, but it’s there. Maybe math works the same way. These constants and structures reflect some hidden architecture of the universe; the relations between distant practical problems shadow a secret connections in the world of mathematics where the two problems get closer to each other. The example of pi seems to reflect a certain kind of symmetry condition, for example. The usefulneds is unreasonable only because we overlooked the underlying structure of the world.

Discovery:
As for discovery, I read the natural speech related to the human cerebral cortex article.It's fascinating to see how principal component analysis can turn some kind of massive data into a particularly high dimensional space and then possibly lower the dimension and create visualizations and help us to see the data in a crystal clear way. Here it sounds still within the realm of linear algebra in mathematics, but in certain dimension cutting linear subspace is not an optimal choice because it might loss too much information and people use manifolds to deal with the situation to lower down the dimension furthermore, but at the cost of inquiring some sophisticated geometry and this is where topological data analysis come in。

[rshah1869]

Scientists build computer models of natural or social phenomena to understand, predict, and manipulate systems that are often too intricate or inaccessible to study directly. As we discussed in week 4, models and distributions use mathematical phenomena to simulate scenarios, test. For instance, climate models simulate atmospheric and oceanic interactions to forecast future climate patterns, while economic models predict market responses to policy changes. In his work The Mathematical Universe, Tegmark discusses how models can reveal the underlying mathematical structure of physical reality, highlighting their role in making sense of complex systems that may not be directly observable.

One major advantage of computational models is their ability to conduct experiments that would be impractical or unethical in the real world. For example, models can simulate disease spread without exposing populations to risk, or predict the impact of environmental policies without disrupting ecosystems. Additionally, models are essential when dealing with phenomena that occur over vast spatial or temporal scales, like galaxy formation or climate change. This flexibility makes models indispensable tools in fields ranging from physics to social science.

To determine whether a model is “good enough” to learn from, scientists perform validation by comparing model predictions with real-world data. If the model consistently produces accurate results within acceptable error margins, it is considered reliable for its intended purpose. Sensitivity analysis also helps identify which variables most influence outcomes, enhancing model robustness. When using these kinds of models however, we must remember that they are simplifications. Even highly accurate models may fail when applied to new or unanticipated conditions.
Models and theories are related but distinct in science. A theory offers a broad explanatory framework, while a model operationalizes that theory into a form that can be tested and refined. Models translate theoretical concepts into computational or mathematical structures, allowing for hypothesis testing and prediction. For example, models of neural activity in Huth et al.’s study on brain function take theoretical ideas about semantic processing and use data-driven methods to map how these processes manifest in the human cortex. In Sanchez-Lengeling and Aspuru-Guzik’s paper, machine learning is a major component of “inverse design,” which aims to discover tailored materials from the starting point of a particular desired functionality.

When two equally strong but distinct models coexist in a field, scientists often compare their predictive accuracy and theoretical coherence. Sometimes, one model may be favored based on empirical validation, while in other cases, both models may coexist if they capture different aspects of the phenomenon. For instance, classical and quantum models describe physical systems differently but are used complementarily depending on the context.
Computational models are essential for exploring complex phenomena. They offer insights that pure theory or direct experimentation cannot achieve, and are constantly evolving, shaped by new data and theoretical advancements, serving as bridges between abstract concepts and real-world applications.

Discovery:
I read the Tegmark paper where he described the competing hypothesis that seek to explain the mathematical universe. Tegmark’s argument challenges the conventional view that models merely approximate the real world. Instead, he proposes that if the universe itself is a mathematical structure, then models are reality rather than just representations of it. Scientists often favor models that are symmetrical and mathematically elegant because these properties tend to align with physical reality. Tegmark suggests that this preference is not coincidental but rooted in the idea that reality itself is fundamentally mathematical. Therefore, the elegance of a model might reflect a deeper, inherent symmetry in the universe. This is not a mainstream idea as currently in science and statistics, the prevailing idea is that mathematical models merely approximate reality. I wonder what the world might look like and what the development of science would be if we collectively believed in the Mathematical Universe Hypothesis.

[Noah48841]

I believe that scientists largely build computer models of social or natural phenomena to be able to predict future behavior (largely to then make claims about how we should make decisions) and explain past behavior. The Monte Carlo Method is pretty good evidence of this theory, I believe, as it doesn’t reflect something true about the universe (as scientists often say that they care about truth for its own sake, which I don’t think is economically sustainable nor do I think is the fundamental drive here), but a distribution over likely scenarios.
Scientists can, according to Wigner, to a whole lot with the models – even to the degree for which this is surprising. You can do anything from talk about the most basic stuff we know about the universe (particle physics and quantum mechanics) to modelling organisms (biology) to populations of organisms (evolution and ecology) to organizations of humans (economics) and even what we value/ how we should think/ acting under uncertainty (formal ethics/ formal epistemology/ decision theory). Most subjects that can be thought of as a rigorous enough subject to actually be discussed can probably be mathematically modelled in some way or another.
This is a great question! I think that there are often two ways we think about theories (maybe there are more, and I haven’t thought about this very much): 1) heuristics of a model (i.e. economic theory of supply and demand) or 2) the rules underlying some model (i.e. Grand Unified Theory). However, we often use this word without indicating which term is being used - though this is often not important because context is clear or that we don’t know if some theory is a heuristic or the actual underlying rule of some system. A model, on the other hand, is an abstracted version of a system that keeps the important parts that we want to study (i.e getting rid of noise).
I think this last question is one of Kuhnian paradigms – usually, we want to say, when there are multiple paradigms at play, that one is more correct or that neither are correct. To say that two models that predict different phenomena are correct is to just accept the contradiction. To say that two models predict the same phenomena and just have different interpretations (say, of the metaphysics, ontology, etc) seems merely semantic.

DISCOVERY:
I think I learned quite a lot about the political history of climate change – particularly about the influence of global models. This is particularly important to me as I am a part of a movement called Effective Altruism, which believes, and I, for the most part, agree, that society greatly underrates long termism and along with that threats from existential risks like AI, nuclear, and authoritarian value lock in. It is therefore important to know how ideas that started out out-there and had no immediate cost became palatable among politicians and the public.

[mhmanek]

This week’s exploration begins with a simple question: why do scientists build computer models of natural or social phenomena? Beneath the hum of processors and the glow of screens lies a deeper impulse—to craft a “digital double” that captures not only what we observe, but also what we cannot directly perceive. From the earliest Monte Carlo experiments, which embraced uncertainty rather than denying it, to Wigner’s meditation on mathematics’ uncanny fit with nature, models have coevolved with computation to become more than mere calculations. In climate science, as Edwards shows, models serve as our only window into Earth’s future, compressing millennia of atmospheric complexity into a form we can interrogate and, in doing so, invent new insights.

What can scientists and engineers do with models that they cannot—or may not—be able to do with the phenomena themselves? In silico experiments grant freedoms that physical reality denies: the ability to rewind, accelerate, amplify or mute every parameter at will. Engineers test bridge designs under impossible loads, epidemiologists simulate outbreaks in virtual cities, and chemists explore molecule formations light-years faster than any lab synthesis. These digital doubles become laboratories without walls, letting us expose hidden vulnerabilities, probe “what-if” scenarios at planetary scale, and safely explore corners of possibility that would be too costly, dangerous or slow in the real world.

Determining whether a model is “good enough” requires its own kind of experimental rigor. Calibration against observational data establishes a first line of credibility, but true confidence emerges from rigorous validation, sensitivity analysis and the humility to recognize gaps. A climate model that reproduces past temperature trends is necessary but not sufficient—one must also stress it against extreme inputs, cross-check it with independent datasets and observe whether small tweaks unravel its predictions. Only when systematic errors fall below tolerances—set by the decision-makers who will rely on its forecasts—can we trust a model to teach us something beyond its own assumptions.

At the heart of this practice lies the question of how a model relates to theory. A theory provides the abstract scaffolding—laws, equations, principles—while a model breathes life into those abstractions through specific algorithms, parameter choices and numerical schemes. They are not the same: a theory without a model remains inert and untested, while a model without theory is a black box of uncertain provenance. Together, they form a virtuous cycle: theory guides model design, and model output refines theory, each feeding the other in a dynamic exchange.

Finally, what becomes of progress when two—or more—equally strong yet distinct models vie for acceptance? Scientific communities often push toward convergence, driven by the desire for a shared framework that eases communication and coordination. Yet divergent models can coexist fruitfully, providing cross-validation, revealing blind spots and spawning hybrid approaches. When disagreements persist, the tension itself becomes a catalyst for discovery, forcing practitioners to question hidden assumptions and ultimately forge richer, more resilient representations of the world.

Discovery
I find it interesting how emergent systems differ from fundamental systems in this regard - in physics (fundamental), a mathematical model is a detached way of referring to/describing the physical behaviors - but in say the stock market (emergent), a mathematical model is no longer detached but essentially tied to the system being described, since trades can be made based on those models (hence the phenomena like Flash Crash, i.e. stock market crashes due to algorithmic trading).

[eeengland1]

The idea of implementing computer simulations and the Monte Carlo method to solve differential equations without the need for explicitly finding solutions is remarkable and intriguing. Why try to model quantum mechanics or particle physics when you can simulate it through relative randomness and achieve a desirable outcome? The Monte Carlo method, in this sense, most likely encouraged the use of computers for a wide variety of brute force and computationally costly tasks at less work for the individual, throughout many fields of science.

Somewhat related: weather simulations, run on supercomputers that take a relatively small input and run with synthesized data to predict regional or global atmospheric patterns and conditions is a remarkable feat of computer science and physical science theory. Humans have even learned from our simulations, using computer models to build mathematical ones, and climate simulations, while far more complex than most other computing applications, also seems like a field that could have benefited in the early stages through the Monte Carlo method.

However, computer models still leave much to be desired. For one, when constructing human models, LLMs lack the experience that real humans do in physical social interactions, and sometimes prioritize responses that share the most common opinions of society instead of expressing a more unique range of beliefs and worldviews. Furthermore and in extension, LLMs still lack diversity of thinking. While computer models and simulations might be effective at modeling physical systems in the absence of hard mathematical proofs, they seem less optimized for human tasks.

But mathematics has always been an integral part of the sciences from the beginning. Wigner’s pondering on the seemingly specially-built mathematical logic that is directly applied to the sciences brings with it a number of notable questions that highlight our lack of comprehension, for one, why are the two fields so intricately tied? Or can other mathematical frameworks just as easily describe our world?

Discovery: The concept of a digital double is an odd one, something that I don’t think is an inevitable result of the advent of competent AI but instead a solution to our socio-logistical problems, and something that I believe may not catch on. The idea of removing yourself from a social interaction meant for you in place of an artificial intelligence modeled after you could certainly rest uncomfortably in the minds of many, especially since correspondence between the AI and the person in question would have to occur after every AI-taken interaction, to keep the human up to speed on what “they” did. Furthermore, developing AI systems that contain enough information about you to simulate you perfectly is concerning to many from a security standpoint, after all, numerous issues could arise from a stunt double performing actions that may be against your actual wishes.

[ciara-kinney]

When considering this subject of “digital doubles, I couldn’t help but be reminded of Queletet’s l’homme moyen. This seemed particularly relevant in using AI agents in sociological studies. If these models are developed by syntheiszing a huge amount of data into one individual agent, which contains all this information and appropriate weighting, then they, in my understanding, represent the average of the data you’ve input, with whatever skewing you want to apply to it. Putting an agent representing a rural farmer and an agent representing an urban environmental activist in conversation thus seems analogous to putting l’rural-farmer moyen and l’urban-environmental-activist moyen in conversation with each other.

In a way, this seems to be just shifting the work of averaging to a different part of the study– instead of sampling conversations between numerous individual environmental activists and farmers and then averaging the content of these conversations in analysis afterwards to create meaning, the averaging is front-loaded, and only a single conversation that processes all this data needs to be conducted. Of course, there is the question of what data is being lost in either process, which was discussed to some extent, but I suspect that the nuances of any particular case are too great to every really capture. But again, if a few insubstantial nuances are lost, how much effect does this really have if they are sufficiently small and this new system is sufficiently more effective?

I also found Paul Edwards’ "A vast machine" very interesting. He discussed the process of creating simulations for complex systems, focusing on climate models. The need to reach back into the past to retrieve data complicates the process hugely, when new data cannot be generated without a huge amount of time. The “friction” of both the data and the meta data cannot be understated, and these projects are constrained by such complications. There are also issues with data that is not recorded, and cannot be easily fit into the model such as “housework, unpaid child labor, [and] barter” which Edwards mentions in reference to building a model of a global economy. These activities are not quite as naturally described in monetary terms, so one encounters not only the issue of lacking adequate data, but also that of confronting important data which must be distorted to some degree to fit within the model you are evaluating. I’m sure that it is best to keep a close eye on both whatever model you are constructing, and the real system that you are simulating.

Discovery: For my discovery this week, I read some of the responses to Wigner’s Unreasonable Effectiveness of Mathematics in the Natural Sciences, specifically Hamming’s and Grattan-Guinness’. I found the discussion very interesting. I found myself more convinced by Hamming’s argument that mathematics and physics have coevolved and this offers some explanation for mathematics’ remarkable applicability. However, I don’t think that this answers the question of how remarkable it is that our world can be described very precisely in this language– ie that the physical world obeys consistent rules at all. I really liked the quote: “It is not “natural” that laws of nature exist, much less that we can discover them”.

[TonyYZ]

Response
Scientists use mathematics both as a conceptual and quantitative tool to describe subsets of phenomena in the world consistently. The concept of modelling and simulation arose from this historical context, which differs from theoretical constructions thanks to the accessibility to computational resources. As Wigner compares “the laws of nature” with “the construction of machines” in his speech, scientists attempt to reduce the world into a machine-like system with controllable parameters (12). However, since the range of controlled variables is limited, there is always a gap between developing models and genuinely copying the system. This gap reflects Wigner’s “unreasonable effectiveness” paradox – mathematical structures somehow capture essential features of nature in ways that exceed our expectations.
While this gap means inevitably misunderstanding parts of the system, models capture certain mechanisms abstracted from the original system by avoiding the costs of reconstructing natural systems with high-level complexity. The Monte-Carlo method allows the computation of neutron diffusion by randomly generating “neutrons” represented by “80-entry punched computer card[s]” (Eckhardt 133). The simplification comes from replacing the mechanistic behaviors of neutrons with random choices from a possible set of such behaviors. Since macro-level phenomena are not sensitive to specific micro-level events, random patterns that satisfy distribution and regularity standards preserve macro-level properties with low costs. As Metropolis and Ulam note, these statistical approaches enable scientists to investigate phenomena otherwise “impractical or impossible to explore” through direct experimentation (336).
Models also allow experimentation with different parameters. Earth’s atmosphere can be simulated with General Circulation Models (GCMs). By changing atmospheric parameters, scientists use GCMs to estimate temperature changes resulting from influences like doubled CO2 levels (Edwards 376). This virtual experimentation represents what Edwards calls “working in model worlds” – creating computational spaces where scientists manipulate variables beyond what would be possible in reality (14). However, models require constant incorporation of new parameters to address more general phenomena. Climate models failed to reach the “nuclear winter hypothesis” until “urban and forest fires” and “the staggering quantities of smoke” were included (Edwards 381).
Raup’s work on shell coiling demonstrates another modeling power – generating hypothetical forms that may not exist in nature. By simulating shell morphology through four geometric parameters, Raup produced “the total spectrum of physically possible forms” as a consequence of the interactions between constraints of biological forms and unexplored possibilities (1178).
The multiplicity of climate models illustrates a final aspect of scientific modeling – productive tension between competing models. Rather than converging toward a single “correct” model, climate science has developed ensemble approaches that embrace model diversity, acknowledging that models are always partial representations shaped by their creators’ assumptions, similar to how Kozlowski and Evans observe that computational models inevitably embed perspectives from their human designers.

Discovery
I find the alienness of LLMs interesting. For example, Chat-GPT does not have much of a problem in deciphering base-64 strings, as the corpus it is trained on has a lot of such strings. The training process requires the model to learn to translate from base-64 to English, just like it knows how to translate from German to French, but in a much more mechanistic version. Interestingly, it seems that humans learn languages more naturally than code like base-64 and Morse code, yet GPT uses the same learning process regarding the two types of symbolic systems. If we designed a way to convert the base-64 string into a pronounceable string of phonemes, it might be possible to train humans to do the same. However, it shows humans’ reliance on the vocal organs, which is not addressed by the design of GPT. How can we incorporate this aspect into its design if we want more realistic replacements for human participants in sociological research?