Friday, June 21, 2024

The use of random procedures in describing non-random aspects of the world

 It has often been argued that quantum mechanics is no help to support free will because, although it does deny absolute determinism, it only puts random occurrence of events as an alternative, and randomness cannot support the idea of free will. This thought is usually based on an assumption that randomness is somehow without meaning, without significance, without reference to anything substantially real. The two articles below I think make that assumption untenable.


The Counterintuitive Power of Randomness



Sometimes an idea’s time arrives. In the late 1940s, the idea that randomness can be a powerful tool arrived in New Jersey. Claude Shannon, an electrical engineer at the Bell Telephone Laboratories in Murray Hill, used random codes to show that it is possible to transmit information over arbitrarily noisy channels. Independently, Paul Erdős, a Hungarian mathematician working at the Institute for Advanced Study, some 30 miles to the southwest, used arguments depending on randomness to prove a seminal result in graph theory. Mathematical graphs are collections of points with lines connecting some of them. Erdős showed that in small enough graphs, it’s possible to avoid creating groups of points, called cliques, that are all connected or all disconnected. 

In physics, the role of randomness had undergone a profound change during the previous generation, as the discovery of quantum mechanics indicated that the universe is inherently random. But while physicists saw randomness as a challenge to be overcome, mathematicians and engineers figured out how to put it to use. As Rahul Santhanam of the University of Oxford explained, there’s something paradoxical about the way randomness helps mathematicians solve problems.

They often use it to prove that a given mathematical structure exists without specifying how to build it. For example, Erdős’ proof of the existence of clique-less graphs didn’t specify how to make them. Instead, he showed that if you consider the set of all possible graphs of a given size, and choose one at random, the chance that you’ll find a graph without a “forbidden” clique is greater than zero. Which means that such a graph must exist.

Shannon’s result was the beginning of a discipline called information theory, which explores how much information can theoretically be transmitted in particular circumstances — though it does not necessarily spell out the best way to do so. Like Erdős, Shannon didn’t specify how to create a scheme for reliable transmission over a noisy channel. But, using randomness, he proved that such a way must exist.

Since then, mathematicians have used randomness as a tool not just in graph theory and information theory, but also in geometry, analysis (an advanced form of calculus), combinatorics (the study of counting methods) and computer science. Earlier this year, Avi Wigderson of the Institute for Advanced Study won a Turing award, one of the top honors in computer science, in part for his work studying connections between randomness and computation.

In recent years, mathematicians have been probing the limits of probabilistic methods and gaining intuition for where they might fail. “It’s very, very natural to try to use randomness to try to push things through,” one mathematician told me. However, “randomness only gets you so far.” 


What’s New and Noteworthy

Still, it gets you very, very far. Researchers have continued to follow in Erdős’ footsteps, using randomness to prove many results in an area called Ramsey theory, which studies the unavoidable formation of cliques in graphs. In 2020, for instance, two mathematicians improved the lower bound on numbers that quantify how big a graph must get before certain patterns become inevitable. The following year, Quanta reported on a proof that marked major progress toward resolving one of the oldest problems in Ramsey theory, a question about how long disordered strings can be. And last year, I wrote about yet another Ramsey result where randomness was crucial.

Probabilistic methods have also made it possible to prove the existence of other kinds of structures. In 2017, mathematicians heaped one random process on top of another, only to find that consistent geometric patterns arise amid all that disorder. “The disorder converges to a universal form,” wrote Kevin Hartnett. “At the precise moment when a random system seems most chaotic, exquisite geometric order peers through.” Last year, I wrote about how mathematicians used a probabilistic method to prove that there are infinitely many configurations known as subspace designs — objects related to error-correcting codes — “whose existence is not at all obvious,” as one mathematician told me.

In all this work, mathematicians have to be clever about how they employ randomness. In 2022, for instance, I covered a groundbreaking proof of the Kahn-Kalai conjecture, a major problem that asked when phase transitions occur in graphs and other systems. Two mathematicians gave the answer by randomly selecting pieces of graphs and sets until they gradually built up the structures they needed, rather than applying randomness in one fell swoop.

Randomness seems like the very antithesis of everything math purports to be — the steadfast pursuit of logic, the search for patterns and structure, the crafting of neat and airtight arguments. And yet it’s become one of the subject’s most useful instruments.





Nature Reviews Immunology

Why the immune system takes its chances with randomness

Nature Reviews Immunology volume 14page711 (2014)Cite this article

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In their recent Opinion article (Lymphocyte fate specification as a deterministic but highly plastic process. NatureRev.Immunol., Reiner and Adams presented a fascinating deterministic interpretation of how lymphocytes acquire different fates. They propose that the generation of multiple lymphocyte subsets from each precursor occurs via an inevitable developmental pathway. This deduction is based on the premise that the system is too important to be left to stochastic processes. To account for recent evidence to the contrary, stochastic processes are suggested to only appear under conditions in which artificially large numbers of responding precursors might relax the deterministic programme (as used in Refs 2,3) or under in vitro conditions in which the usual three-dimensional (3D) arrangement of externally delivered signals that channel fates is removed (as used in Ref. 4). In other words, stochastic mechanisms only occur when experimental conditions happen to support the role of randomness.

There are, however, several reasons — as outlined below — to challenge the premise that stochastic processes are not equally up to the task of generating a reliable immune response.


The authors themselves point out that evolution exploits randomness for the most important task of all — creating lymphocyte receptor diversity. Other immune examples of stochastic processes include the probabilistic expression of cytokines5,6 and the combinatorial expression of natural killer cell receptors in a population7.


In the imagined B cell and T cell odysseys1, at least six intricate moves must take place to generate the different cell fates. A distinct deterministic pathway is needed for each, and the correct set of signals must be received in the correct order by each of potentially thousands of progeny; lymphocytes and numerous other cells must encode complex instructions for orchestrating the right set of signals to generate every cell type at the right time. By contrast, by using stochastic processes multiple cell types can be generated with much simpler instructions4,8,9,10,11, even in the absence of environmental direction.


It is tempting to observe the complex structures and cell interactions of primary lymphoid tissue and deduce that they are crucial for the formation of heterogeneous outcomes. This hypothesis has been tested by asking what remains when such structures are removed. We and others find a great deal of cell fate heterogeneity under simple in vitro culture conditions4,5,6,12,13. Conversely, crucial molecular contributors to early developmental programmes, including asymmetric cell division, do not alter B cell or T cell responses in vivo14. Thus, although the 3D environment and asymmetric programming might have some role in modifying cell fate allocation, they are not the only sources of variation.


In the stochastic interpretation, variation is inherent and consistent immune outcomes only arise when considering the population as a whole. As Reiner and Adams point out, the number of antigen-specific precursors recruited into the immune response is a crucial variable, and may be as low as 20. However, mathematical models in which randomness drives cell fate selection suggest that a reasonably robust immune response can be achieved even with starting cell numbers of this order4,9,10,15. Thus, a role for randomness should not be rejected on this basis alone.


As a research community, we have not yet acquired all of the data required to answer how both deterministic and stochastic processes interleave to build the complete immune response. However, along with Reiner and Adams, we look forward to the resolution of this conundrum. Perhaps unlike them, however, we are gamblers, suspecting that the immune system does play a game of chance, albeit with the rules having evolved so that the odds are stacked in our favour.


  1. Reiner, S. L. & Adams, W. C. Lymphocyte fate specification as a deterministic but highly plastic process. Nature Rev. Immunol. (2014).
  2. Buchholz, V. R. et al. Disparate individual fates compose robust CD8+ T cell immunity. Science 340, 630–635 (2013).
  1. Gerlach, C. et al. Heterogeneous differentiation patterns of individual CD8+ T cells. Science 340, 635–639 (2013).
  1. Duffy, K. R. et al. Activation-induced B cell fates are selected by intracellular stochastic competition. Science 335, 338–341 (2012).
  1. Guo, L., Hu-Li, J. & Paul, W. E. Probabilistic regulation in TH2 cells accounts for monoallelic expression of IL-4 and IL-13. Immunity 23, 89–99 (2005).
  1. Kelso, A., Groves, P., Troutt, A. B. & Francis, K. Evidence for the stochastic acquisition of cytokine profile by CD4+ T cells activated in a T helper type 2-like response in vivoEur. J. Immunol. 25, 1168–1175 (1995).
  1. Raulet, D. H. et al. Specificity, tolerance and developmental regulation of natural killer cells defined by expression of class I-specific Ly49 receptors. Immunol. Rev. 155, 41–52 (1997).
  1. Rohr, J. C., Gerlach, C., Kok, L. & Schumacher, T. N. Single cell behavior in T cell differentiation. Trends Immunol. 35, 170–177 (2014).
  1. Duffy, K. R. & Hodgkin, P. D. Intracellular competition for fates in the immune system. Trends Cell Biol. 22, 457–464 (2012).
  1. Subramanian, V. G., Duffy, K. R., Turner, M. L. & Hodgkin, P. D. Determining the expected variability of immune responses using the cyton model. J. Math. Biol. 56, 861–892 (2008).
  1. Hodgkin, P. D. A probabilistic view of immunology: drawing parallels with physics. Immunol. Cell Biol. 85, 295–299 (2007).
  1. Hasbold, J., Corcoran, L. M., Tarlinton, D. M., Tangye, S. G. & Hodgkin, P. D. Evidence from the generation of immunoglobulin G-secreting cells that stochastic mechanisms regulate lymphocyte differentiation. Nature Immunol. 5, 55–63 (2004).
  1. Bird, J. J. et al. Helper T cell differentiation is controlled by the cell cycle. Immunity 9, 229–237 (1998).
  1. Hawkins, E. D. et al. Regulation of asymmetric cell division and polarity by Scribble is not required for humoral immunity. Nature Commun. 4, 1801 (2013).
  1. Duffy, K. R. & Subramanian, V. G. On the impact of correlation between collaterally consanguineous cells on lymphocyte population dynamics. J. Math. Biol. 59, 255–285 (2009).


P.D.H. and M.R.D. are supported by National Health and Medical Research Council (NHMRC) Fellowships and NHMRC grants 1057831 and 1054925, and Independent Research Institutes Infrastructure Support Scheme Grant 361646. K.R.D. is supported by Science Foundation Ireland grant 12 IP 1263. P.D.H. and K.R.D. are also supported by the Human Frontier Science Program, grant RGP0060/2012.

Author information

Authors and Affiliations

  1. and the Department of Medical Biology, Walter and Eliza Hall Institute of Medical Research, Parkville, Melbourne, 3052 Victoria, Australia, University of Melbourne, Melbourne, 3010 Victoria, Australia.,

Philip D. Hodgkin & Mark R. Dowling

  1. Hamilton Institute, National University of Ireland, Maynooth, County Kildare, Ireland

Ken R. Duffy

Corresponding author

Correspondence to Philip D. Hodgkin.