• Apache Mahout - a highly scalable set of machine learning libraries. Implements common text-mining algorithms, including LDA, SVD and various clustering methods. Java only. Related to Lucene and Hadoop.
• Carrot2 - open source search results clustering engine. Implements the STC and Lingo algorithms along with components for retrieving results from common search engines. Supports Java and .Net
• GATE - Java-based open source software capable of solving almost any text processing problem...the Eclipse of Natural Language Engineering, the Lucene of Information Extraction, the ISO 9001 of Text Mining
• GenSim - free Python framework designed to automatically extract semantic topics from documents, as efficiently (computer-wise) and painlessly (human-wise) as possible...Latent Semantic Analysis, Latent Dirichlet Allocation or Random Projections, discover semantic structure of documents, by examining word statistical co-occurrence patterns within a corpus of training documents
• Katoa - a toolkit for concept-based text processing. Katoais a Maori word meaning everything, and stands for Knowledge Assisted Text Organization Algorithms
• Lemur - an ongoing open source project developing search engines (Indri), text analysis and query logging tools. Supports Java and C++.
• LingPipe - tool kit for text analysis tasks named-entity extraction, classifying Twitter results, query spelling correction. Limited free version available for non-commercial use.
• Lucene - a high-performance, full-featured text search engine library written entirely in Java. Suitable for nearly any application that requires full-text search, especially cross-platform. Open source. Wrappers/ports for .NET and Python.
• MALLET - a Java-based package for statistical natural language processing, document classification, clustering, topic modeling, information extraction, and other machine learning applications to text
• NaCTeM - a long-term project providing access to a range of text mining tools and services (consultancy, tutorials, test corpora)
• NLTK - a suite of libraries and programs for symbolic and statistical natural language processing (NLP) for the Python programming language
• OpenNLP - is a machine learning based toolkit for the processing of natural language text.
• RapidMiner - is a popular, open source, integrated platform for data/text mining and visualization. Based on WEKA, but not dedicated to text mining. GUI for end-users, CLI for server-side, and Java API for developers.
• SentiStrength estimates the strength of positive and negative sentiment in short texts, even for informal language. It has human-level accuracy for short social web texts in English, except political texts. Web form test plus .Net and Java versions available. 
• Stanford NLP tools - statistical NLP toolkits for various major computational linguistics problems, written in Java
• TinyCC - text corpus production engine that can be used to produce corpora in Leipzig Corpus Collection (LCC) format.
• Terrier - an open source search engine that implements state-of-the-art indexing and retrieval functionalities, and provides an ideal platform for the rapid development and evaluation of large-scale retrieval applications
• TextTrend - an academic project developing an integrated text-mining and network analysis toolbox, with a particular emphasis on temporal analyses. See use cases for examples of applications. Related to CIS Shell project.
• tm - is an open source text-mining infrastructure/framework for R users
• UIMA (Unstructured Information Management Architecture) - an open, industrial-strength, scaleable and extensible platform for creating, integrating and deploying unstructured information management solutions from powerful text or multi-modal analysis and search components. Comprises frameworks (for Java, C++), components and infrastructure. Conceived by IBM, now an Apache project.
• WEKA - a collection of machine learning algorithms for data mining tasks. The algorithms can either be applied directly to a dataset or called from your own Java code. Contains tools for data pre-processing, classification, regression, clustering, association rules, and visualization.
• WordNet - is the definitive lexical database of English. Nouns, verbs, adjectives and adverbs are grouped into sets of cognitive synonyms (synsets), each expressing a distinct concept

Visualization

• Birdeye - an open source information visualization and visual analytics library for Adobe Flex (Flash).

 

Maths

• ALGLIB - a cross-platform, open source numerical analysis and data processing library. It supports several programming languages (C++, .Net, Pascal, VB, Python) and several operating systems (Windows, Linux, Solaris).
• Eigen - an open source C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms. Highly optimised to make effective use of caches and SIMD extension sets. Seems single threaded for now but parallelisation is planned.
• Math.Net Numerics - an open source library for the .NET Framework and Mono, providing a framework of numerical-scientific data structures and algorithms. Supports dense/sparse linear algebra and statistical functions and is parallelised in parts to improve performance
• SciPy - an open source library of scientific tools for Python. It depends on the NumPy library, and it gathers a variety of high level science and engineering modules together as a single package, including statistics and linear algebra
• Sylvester - vector and matrix maths library for Javascript. Interesting project for those interested in browser applications.