A Digital History of Philosophy of Mathematics

Max Noichl, Gareth R. Pearce & Charles H. Pence

2024-08-22

Outline

  • What we are trying to do?
  • Problems!
    • Best approach to modeling philosophical texts?
    • Identifying a sample?
    • Cleaning up texts.
  • Some results

What we are trying to do

  • Digital large-scale history of a philosophical field, philosophy of mathematics (PoM).
  • Identify prefigurations of later trends, especially wrt. to philosophy mathematical practice.
  • Very much WIP!

Partition a corpus and show its historical development. Graphic: Malaterre, Pulizzotto, and Lareau (2020)

How to model philosophical texts

  • How to represent texts? – Topic modelling ran for us into all kinds of problems!
  • BOW-inherited problems - Polysemy, collocation, etc.
  • Model identifiability, Unbalanced Corpora? (Agrawal, Fu, and Menzies (2018),Veselova and Vorontsov (2020))
  • Heavy information-loss before further processing.
  • These problems might get worse, if you expect only subtle topical differences.
  • Answer: Transformers! (BERT, RoBerta, Deberta… – nomic-embed Nussbaum et al. (2024))

New Problem!

Out of the box models don’t know much about philosophy.

Answer: Fine-tune!

How to identify PoM

  • There is no grand old journal Philosophy of Mathematics!
  • Philosophy of Science has been running since 1934. Earliest dedicated PoM journal started in the mid 90s…
  • PoM is spread out over dozens of more general journals
  • Similar, but distinct literatures (philosophy of physics, philosophical logic)
  • This is a PU (Positive - Unlabeled) learning problem

The Data

  • Large base-sample size: JStor + WOS (283,744 items after filtering, 400k+ before).
  • Refined by journal tags & PhilPapers journal list.
  • Time range: 1900-2021.
  • Interactive data-cleaning strategy.
  • Philosophia Mathematica (Series 2 & 3): 840 items

Sample-Distribution

A lot of older texts are messy in pretty unforseable ways. How to do cleanup? One answer: Unsupervised learning + interactive data-selection.

Alternative clean-up approach:

Segmentation with PleIAs

PleIAs-playground-data

This is approximately the length of a generation, but the factors that made for creativity at this time are
not connected with a particular stage of the individual life-cycle: at the height of the movement in the
1790s, Kant was a creative oldster in his 70s, while Schelling was a Wunderkind of barely 20; Fichte,
Hegel, Schopenhauer and other figures did their creative work at ages varying from early adulthood to
late middle age. It is as if tese individuals, at various times in their life spans, intersected with a
situation of cultural capital, market opportunities, and resulting emotional energies; whatever their
This content downloaded from 131.130.169.5 on Fri, 14 Apr 2017 21:32:01 UTC All use subject to http://about.jstor.org/term
THEORY OF INTELLECTUAL CREATIVITY
personal ages, those who were conncted in the right way had a creative upsurge, that lasted as long as
those conditions lasted. It has often been noticed that creativity clusters this way in particular “Golden
Ages.” The usual macro explanations, though, are weak. One popular formulation is to refer to the
“Zeitgeist,” the spirit of the age; but this only labels the problem, not explains it. A more penetrating
hypothesis is a macro-reductionism, like that of Marxians and others, which asserts that ideas reflect the
larger structure of the society, especialy its mode of economic production. But this is plainly inadequate.
What is distinctive about the ideas produced by intellectuals is that they come from a specialized
community, based on institutions which give it the autonomy to break away from the discourse of
everyday life. On the other hand, intellectual communities do have a historical location in the larger
macro-structure of society. The best theory should interate several layers of causation: from the largest
macro structures, on down to the micro experiences of intellectuals’ own creativity. 
Multi-level Determinism 
It is possible to study the internal structure of the intellectual world as the locs of a
two-fold social causation of ideas: on one level, there is a

PU learning with Spies

  • Train XGBoost (Hyperopt with Parzen Estimator, 3-fold cross.-val.) on a mix of [unlabeled data + spies] as negative class and the remaining positive class.
  • Use the trained model to predict sample-likelihoods. Choose a threshold such that the spies are separated well. This leads to N (likely negative group with some spies) and U (unlabeled group, containing most of the spies).
  • Use N as negative and P as positive group to train new classifier. Use it as your final model.
  • Following tamosiunas (2023), Liu et al. (2002)

UMAP of PM (yellow), and selected PoM (red) within the whole corpus. (Only subsample shown: n=10000)

Distribution of selected PoM (red) over journals in the whole corpus.

Tentative Results

Further Work: Technical

  • Comparison of different LLMs
  • Fine-tunig infrastructure is acceptable, but the tuning itself somewhat brittle.
  • Actual cross-temporal analysis.
  • Adjust and justify clustering parameters.

Further Work: Philosophical

  • Test specific hypotheses in the HoPoM (Mancosu (2008))
  • Identify overlooked contributions to the Philosophy of Mathematics
  • Apply these methods outside of PoM

Thank you!

Literature

Agrawal, Amritanshu, Wei Fu, and Tim Menzies. 2018. “What Is Wrong with Topic Modeling? (And How to Fix It Using Search-based Software Engineering).” Information and Software Technology 98 (June): 74–88. https://doi.org/10.1016/j.infsof.2018.02.005.
Liu, Bing, Wee Sun Lee, Philip S. Yu, and Xiaoli Li. 2002. “Partially Supervised Classification of Text Documents.” In Proceedings of the Nineteenth International Conference on Machine Learning, 387–94.
Malaterre, Christophe, Davide Pulizzotto, and Francis Lareau. 2020. “Revisiting Three Decades of Biology and Philosophy: A Computational Topic-Modeling Perspective.” Biology & Philosophy 35 (1): 5. https://doi.org/10.1007/s10539-019-9729-4.
Mancosu, Paolo. 2008. The Philosophy of Mathematical Practice. Oxford: Oxford University Press.
Nussbaum, Zach, John X. Morris, Brandon Duderstadt, and Andriy Mulyar. 2024. “Nomic Embed: Training a Reproducible Long Context Text Embedder.” arXiv. https://arxiv.org/abs/2402.01613.
tamosiunas, rokas. 2023. “Trokas/Pu_learning.”
Veselova, Eugeniia, and Konstantin Vorontsov. 2020. “Topic Balancing with Additive Regularization of Topic Models.” In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics: Student Research Workshop, edited by Shruti Rijhwani, Jiangming Liu, Yizhong Wang, and Rotem Dror, 59–65. Online: Association for Computational Linguistics. https://doi.org/10.18653/v1/2020.acl-srw.9.