The insight I got from watching this interview is that hallucinations in LLM results are generally statistical because they rely on probabilistic vectors. But statistics apply to populations, not individuals. Everything about an individual is a fact.
Algebraic Mind, The: Integrating Connectionism and Cognitive Science, by Gary Marcus. 2001.
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07/10/2026
Sceptic vs. 'Doomer': How Scared Should We Be of Rogue AI?
By: Jonathan Kay
Quillette: 02/10/2026
Jonathan Kay: Mike invented and designed Pipi and founded Ajabbi.
Gary Marcus:Cognitive scientist Gary Marcus, author of The Algebraic Mind and Taming Silicon Valley and writer of the Substack Marcus on AI
...
Quillette podcast host Jonathan Kay interviews AI expert Gary Marcus about the possibility that a rogue artificial intelligence will turn us all into paper clips.
...
This week, I’m going to tackle a big and important subject—artificial intelligence and the risks it could pose to society.
I’m going to be honest here. I’ve been putting this subject off because it’s so big and so technical that I felt intimidated by it.
But the subject is increasingly dominating the news, and I’m a journalist, so it’s time to face the music.
Maybe you’ve been putting off educating yourself about this subject for the same reason.
If so, we’re in this together. So let’s dive in.
The good news is that my guest this week, Gary Marcus, is a real expert—someone who’s been theorising and writing about AI—including its dangers—for more than a quarter century. He’s a well-known American psychologist, cognitive scientist, and book author, who also runs the popular AI blog, Marcus on AI.
But before I run that interview, which I recorded last week, let’s review some key terms, so that the discussion you’re about to hear makes the maximum amount of sense.
First of all, and I’m grossly simplifying here, AI comes in two big categories.
The first, which you’ll hear Gary often refer to as “Symbolic AI,” basically operates like a very big and very sophisticated version of a conventional computer program, with lots of rules, and decision trees, and if-then statements—just like you remember from middle-school computer class, except much, much bigger.
This kind of AI is great for, say, creating a database, or setting out rigid decision trees for an airline pilot to follow. But it’s really bad at a lot of basic tasks that even small children can do, like, say distinguish a dog from a cat, or a plate from a frisbee.
The second kind of AI, which you’ll hear us refer to as “neural networks,” or machine learning, or deep learning, which are overlapping concepts, is completely different.
This kind of AI, which became ascendant in the early 2010s, isn’t created with human instructions. Rather, these systems teach themselves by reference to massive troves of pre-existing data—including through a technical process you’ll hear me refer to as backpropagation.
By way of example, let’s say you want to create a neural network to identify a cat. Instead of writing thousands of lines of code relating to fur and whiskers and meowing noises, you just show the neural network millions of pictures of cats and millions of pictures of things that aren’t cats; and then instruct the computer to teach itself the difference between the two by a computational process of trial and error.
The idea here is that the neural network will start with a completely useless algorithm, then compare the results to the data set, and then try again with tweaked parameters, teaching itself—though a process of machine learning—to refine its self-created algorithms so that it better aligns with the data you’ve already given it.
The humans aren’t giving the computer rules about what a cat is or isn’t. They’re just providing the data trove and then hitting the go button and the computer teaches itself.
These systems are easy to scale, by supplying them with more data and computing power—which is one of their big selling points to investors.
But the problem is that the iterative, self-constructed nature of these mechanisms means they aren’t grounded by predictable rules, and so can sometimes hallucinate artificial realities—a problem that Gary predicted in a 2001 book. Even something as basic as a simple database of public figures, the sort of thing human-resources department at large companies were creating on 1970s-era mainframes, is alien to the neural network information architecture.
As a result, purely neural-network-driven AI systems aren’t very good at following user-supplied rules, even fairly basic ones, like, “write me an academic paper with citations, but make sure those citations aren’t made up,” or “find and download some information for me, but please don’t hack into any proprietary servers in the process,” or, more speculatively, “make a trillion paper clips for me, but don’t make any of them from the bodies of humans.”
I know that last one sounds weird, but I promise it will make sense when you listen to the podcast,
All of this has set off a movement known as the AI doomers, who don’t just think AI will take our jobs and access our banking information—but that it will cause the extinction of humanity itself, by launching nuclear weapons, or creating and unleashing some kind of super powerful bioweapon.
While my guest is not a doomer, he does call himself a sceptic, especially when it comes to what he regards as the allegedly irresponsible and dangerous business practices of certain companies—OpenAI, in particular.
Unlike U.S. senator Bernie Sanders, he doesn’t want to ban hyper-advanced AI systems that greatly surpass human capabilities (which is sometimes called superintelligence). But Gary also doesn’t approve of Donald Trump’s apparently laissez-faire attitude, either.
And he’d like to see a more balanced approach from government, an approach that holds AI companies to account if they don’t place appropriate safeguards on their products, and align those products with human values.
And what would those safeguards look like?
Well, this gets us back into the technical sphere. Gary has long advocated something called neurosymbolic AI, which combines the extraordinary power of neural networks with some of the safeguarding and fact-checking features that can be implemented through the rules-based, predictably algorithmic features of so-called “classic AI”.
It’s all very complicated, I know, but I’m confident I have the guest who can walk us through it.
Please enjoy my interview with Gary Marcus, AI expert extraordinaire.
Working notes on finding a way for Pipi to use genetic algorithms with Hilbert Spaces and matrix calculations. A
desktop exercise with Google for now. Computational experiments on
CPU to come. Some generated Python and CFMLsample code below to start playing with.
It looks promising; it might help address limitations of other methods for using fitness landscapes. The output curves do look better.
This might void the need to use a Cerebras wafer for these fitness landscape calculations.
The next step is to watch free maths lectures on YouTube, then do a lot of drawings and process diagrams. Then play with code and see what happens. 😎
The maths formulas were in LaTeX format and were converted to
non-LaTeX format using the free online LatexConverter.
Update 03/09/2026
Nothing has been verified or tested. 😎
Update 03/10/2026
All OK to use, so it will be added in a future minor release of Pipi. 😎
Making a start on Hilbert Spaces and matrix calculations using CPU
By: Mike Peters & Google Search AI Mode
On a Sandy Beach: 30/08/2026
Mike Peters: Mike invented and designed Pipi and founded Ajabbi.
Google Search AI Mode: Free and surprisingly good.
...
Hilbert Spaces
Hilbert spaces use matrix calculations extensively, especially when the
space is finite-dimensional or when dealing with linear operators. Here is
how they connect:
Finite-Dimensional Spaces
Direct Equivalence: An
(n)-dimensional Hilbert space is
isomorphic to (ℂⁿ) (complex
numbers) or (ℝⁿ) (real
numbers).
Vectors:
Elements in the space are represented as column vectors.
Operators:
Linear operators acting on the space are represented as (n × n)
matrices.
Inner Product:
The inner product is calculated using matrix multiplication
(specifically, the conjugate transpose of a vector multiplied by another
vector).
Infinite-Dimensional Spaces
Infinite Matrices: In infinite-dimensional Hilbert spaces (like L²), the space of square-integrable functions, operators can be
represented as infinite matrices.
Basis Dependence: By choosing an orthonormal basis, any linear
operator can be written as a matrix with infinitely many rows and
columns.
Quantum Mechanics (Matrix Mechanics)
State Vectors: Quantum states are vectors in a Hilbert
space.
Observables: Physical properties (like position, momentum, or
spin) are Hermitian operators.
Heisenberg's Formulation: Werner Heisenberg's formulation of
quantum mechanics is literally called "matrix mechanics" because it uses
matrix calculations to solve Hilbert space problems.
In machine learning, Hilbert spaces—specifically
Reproducible Kernel Hilbert Spaces (RKHS)—rely heavily on matrix
calculations to make complex, non-linear data linearly separable.
Instead of explicitly mapping data points to an infinite-dimensional space,
ML algorithms use the Kernel Trick to compute everything via an
N × N matrix, where N is the number of
data samples.
The Gram (Kernel) Matrix
The core matrix calculation in RKHS is the Gram Matrix (or Kernel Matrix), denoted as K.
Formula:(Kᵢⱼ = k(xᵢ, xⱼ) = ⟨ Φ(xᵢ), Φ(xⱼ) ⟩_{H})
Meaning: Each entry i,j is the inner product of data points
(xᵢ) and
(xⱼ) in the Hilbert space
(H).
Properties: It is always a symmetric and positive semi-definite
(PSD) matrix.
Core Matrix Operations in RKHS
Machine learning algorithms manipulate this kernel matrix using standard
linear algebra:
Matrix Inversion: Used in Kernel Ridge Regression to solve for
weights: (Kv = λ v).
Eigenvalue Decomposition: Used in Kernel PCA to find non-linear
principal components by solving
(Kv = λ v).
Quadratic Programming: Used in Support Vector Machines (SVMs) to
maximise margins using matrix-vector products involving
(K).
Computational Bottlenecks
Because (K) scales with the number of
data points, scaling Hilbert space methods requires specialised matrix
approximations:
Nyström Approximation: Factorises a low-rank sub-matrix to
approximate the massive
N × N matrix.
Random Fourier Features: Avoids the exact matrix calculation
entirely by mapping data into a randomised, lower-dimensional space.
Genetic Algorithms
Combining genetic algorithms (GAs) with Reproducing Kernel Hilbert Spaces
(RKHS) is entirely possible, highly effective, and actively used in machine
learning research.
In this hybrid framework, the fitness landscape is modelled as a
continuous function belonging to an RKHS. This approach addresses a
major limitation of traditional GAs: evaluating an individual's true fitness
can be computationally expensive (e.g., running a physics simulation or a
wet-lab experiment). The RKHS serves as an
intelligent surrogate model.
Core Mathematical Engines At Work
To make this framework highly effective, three core components work in
tandem:
The Representer Theorem: This is the mathematical backbone of your
RKHS surrogate. It guarantees that your continuous fitness approximation
(̂{f}(x)) can be written as a finite linear combination of kernel
evaluations centred at your sampled points:
(ᶠ(x)=∑ ᵢ₌₁ᴺα ᵢK(x,xᵢ))
This scales the search space completely independently of its true
dimensionality, reducing prediction cost to a simple
(O(N)) vector dot product.
Informed Exploration vs. Exploitation: Instead of letting the GA
evaluate individuals blindly on the surrogate, you can use the RKHS
variance (uncertainty) to construct an Acquisition Function (like Expected
Improvement or Upper Confidence Bound). The GA then maximises this
acquisition function rather than the raw estimated fitness, forcing the
algorithm to intelligently search unmapped areas of the landscape.
Gram Matrix Regularisation: As new true data points are evaluated,
they enter the Gram Matrix. To prevent numerical instability or
overfitting as (N) grows, a small
regularisation ridge ((λ I)) is
maintained, smoothly adjusting the landscape's rigidity without rebuilding
the framework from scratch.
How the Combination Works
[GA Population] ---> [Evaluate on RKHS Surrogate] ---> [Select &
Crossover] ^ | |__________________ [Update RKHS with True Data] _________|
The RKHS as the Fitness Landscape: You treat the unknown fitness
landscape (f(x)) as a
smooth function within an RKHS. By evaluating a small set of initial
points, you use Kernel Ridge Regression or Kriging (Gaussian Process
Regression) to build a continuous, analytical approximation of the
landscape.
Genetic Search on the Surrogate: The GA searches this approximated
RKHS landscape. Because evaluating the RKHS kernel matrix is mathematically
cheap (O(N)) for a new prediction, the GA can evolve through thousands of
generations in seconds.
Adaptive Sampling (Bayesian Optimisation): The best candidates
found by the GA are evaluated using the true, expensive fitness function.
These new data points are added back into the Gram matrix, updating the
RKHS landscape to make it more accurate where it matters most.
Key Advantages of This Hybrid Approach
Global Search in Infinite Dimensions: Traditional GAs struggle
with high-dimensional continuous spaces. By mapping the search space into
an RKHS via a kernel (like the Gaussian RBF kernel), the GA
implicitly searches a highly expressive, infinite-dimensional feature
space.
Guaranteed Smoothness: Functions in an RKHS are mathematically
smooth and bounded [6]. This eliminates extreme, discontinuous noise in
the fitness landscape, allowing the GA to find gradient-like paths toward
global optima without getting trapped in microscopic local minima.
Exact Kernel Alignment: You can use a GA to optimise the
hyperparameters of the RKHS kernel itself (e.g., finding the optimal
length-scale (σ ) of an RBF kernel)
to perfectly align the geometry of the Hilbert space with your actual
problem landscape.
Real-World Use Cases
Protein Design & Biochemistry: Designing new molecules where
the "fitness" is binding affinity. The RKHS models the chemical landscape
based on known proteins, and the GA mutates amino acid sequences to find
optimal binders.
Hyperparameter Optimisation: Optimising complex neural network architectures where the
evaluation landscape is highly non-linear and expensive to
compute.
Robotics Control: Evolving locomotion parameters for robotics
where physical testing is slow or dangerous.
Implementation Challenges
The Matrix Bottleneck Returns: As your GA discovers more points
and updates the RKHS landscape, your Gram matrix
(K) grows. Inverting this matrix to
update the landscape scales at O(N³). You will need to use Nyström approximations or sparse
online kernel methods if your total evaluation points exceed 10,000.
Exploration vs. Exploitation: GAs are aggressive exploiters. If
your RKHS surrogate model is inaccurate in an unexplored region, the GA
might get stuck optimising a "fake" peak. You must include an uncertainty
metric (like the variance in a Gaussian Process) to force the GA to
explore unknown areas.
Kernel Ridge Regression or Kriging (Gaussian Process Regression)
Kernel Ridge Regression (KRR) and
Kriging (Gaussian Process Regression, or GPR) are both kernel-based
methods for learning non-linear functions from data, but they differ
fundamentally in their underlying statistical framework, objective
functions, and outputs.
Key Differences
Core Approach: KRR minimises a regularised
mean-squared error loss function in a Reproducing Kernel Hilbert Space (RKHS). GPR uses a probabilistic (Bayesian) approach, defining a Gaussian
process prior over functions and updating it with a likelihood function
based on observed data.
Uncertainty Estimation: KRR outputs point predictions only. GPR
(and Kriging) naturally quantifies uncertainty, providing full posterior
distributions, variance estimates, and confidence intervals.
Hyperparameter Optimisation: KRR typically
optimises kernel parameters using grid search with cross-validation on a
loss function. GPR optimises hyperparameters via gradient ascent on
the marginal likelihood.
Terminology & Origin: Kriging originated in geostatistics
(mining) to find the Best Linear Unbiased Predictor (BLUP), whereas GPR
stems from machine learning and stochastic processes. Mathematically,
standard Kriging is equivalent to GPR under matching covariance and prior
assumptions
When to Use Which
Use Kernel Ridge Regression when: You need a fast, deterministic,
regularised non-linear regressor and do not require predictive uncertainty
bounds.
Use Kriging / Gaussian Process Regression when: You need confidence intervals for predictions, want to optimise
hyperparameters via marginal likelihood, or are modelling
spatial/geostatistical data.
Python Code (not tested)
Here is a clean, modular Python template using scikit-learn to stitch a Genetic Algorithm loop to a Gaussian Process (RKHS surrogate) framework.
import numpy as np
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import Matern
print(f"Current Best Score Found: {np.max(y_train):.4f}\n")
print("Optimization Complete.")
Enhancements
Regularisation (Ridge): The alpha=1e-6 in GaussianProcessRegressor serves as the λI identity-matrix offset. This prevents numerical inversion crashes if the GA proposes points structurally close to existing data.
Normalisation Engine: normalize_y=True transforms unstable fitness functions to a standard normal distribution mapping, stabilising kernel hyperparameter tuning during .fit().
Scalability Hand-off:run_ga_on_surrogate can run for massive populations or generations; it never interacts with true_expensive_fitness, protecting your heavy computation budget.
Script-Only Architecture: Written purely within <cfscript> blocks to remove legacy tag overhead and maximise processing throughput for vector arithmetic loops.
Native Box-Muller Generator: Because ColdFusion’s native rand() produces a flat uniform distribution, a standard Box-Muller transformation was written directly into the mutate() method to correctly inject normal (Gaussian) distribution mutations.
Surrogate Mapping Array Matrix: Replaces Python arrays with native array structs. The predictSurrogateFitness method models continuous field behaviours natively via distances weighted dynamically across the current known points.
Number of data centers worldwide 2025, by country or territory
By:Petroc Taylor
Statistica: 19/11/2025
Petroc Taylor is a researcher with Statista's Technology and
Telecommunications team. His research focus is global developments in the
use of data, including trends in big data, analytics, and storage, as well
as the impact of emerging data technologies across industries and sectors.
He also supports the team's coverage of operating systems,
telecommunications, and the technology industry in Africa..
As of November 2025, there were a reported 4,165 data centers in the United
States, the most of any country worldwide. A further 499 were located in the
United Kingdom, while 487 were located in Germany.
What is a data center?
Data centers are facilities designed to store and compute vast amounts of
data efficiently and securely. Growing in importance amid the rise of cloud
computing and artificial intelligence, data centers form the core
infrastructure powering global digital transformation. Modern data centers
consist of critical computing hardware such as servers, storage systems, and
networking equipment organized into racks, alongside specialized secondary
infrastructure providing power, cooling, and security.
AI data centers
Data centers are vital for artificial intelligence, with the world’s
leading technology companies investing vast sums in new facilities across
the globe. Purpose-built AI data centers provide the immense computing power
required to train the most advanced AI models, as well as to process user
requests in real time, a task known as inference. Increasing attention has
therefore turned to the location of these powerful facilities, as
governments grow more concerned with AI sovereignty. At the same time, rapid
data center expansion has sparked a global debate over resource use,
including land, energy, and water, as modern facilities begin to strain
local infrastructure.
A great introduction to Bayesian Statistics. With example
workbooks. All free, including an Excel Resource Pack. The Real Statistics website by Charles Zaiontz is a fantastic resource.
Update
The YouTube lesson by Vivek Vinushanth Christopher is superb.
Dr. Charles Zaiontz has a PhD in mathematics from Purdue University and
has taught as an Assistant Professor at the University of South Florida
as well as at Cattolica University (Milan and Piacenza) and St. Xavier
College (Milan).
Bayesian statistics uses an approach whereby beliefs are updated based on
data that has been collected. This can be an iterative process, whereby a
prior belief is replaced by a posterior belief based on additional data,
after which the posterior belief becomes a new prior belief to be refined
based on even more data. The initial prior belief in this series may be
based on intuition, previous studies, experience, etc.
In inferential statistics, we commonly test hypotheses, estimate
parameters, and make predictions. In the traditional approach to statistics,
commonly called the frequentist approach, parameters are constants whose
values we aim to discern. Bayesian statistics uses a different approach: we
treat these parameters as variables that have a probability
distribution.
Data Colada is a remarkable effort by Uri Simonsohn, Leif Nelson
and Joe Simmons on topics such as fake data, research design,
meta-analysis, and the reproducibility of science, among others.
